and I will continue always in this track until I shall find something that is certain, or at least, if I can do nothing more, until I shall know with certainty that there is nothing certain

 Here Descartes begin anew his search for something he can know for certain. Remember from Meditation I, he hypothesized that it is at least conceivable that some powerful demon is deceiving him about everything. The remaining question now is whether there is something that  even such a powerful demon could not deceive him.

I suppose, accordingly, that all the things which I see are false (fictitious). . .
To continue the argument, Descartes supposes that his previous hypothesis is correct. Assuming that there were such an evil demon able to deceive in all things, does it follow that one is deceived about everything? Or is there one proposition which, like a the fulcrum point of Archimedes, such that if that proposition turn out to be true, he can deduce other propositions which are true also.
 

Doubtless, then, I exist, since I am deceived . . .

You may have heard the term, "I think, therefore I am" - This famous quote (in Latin, cogito, ergo sum) is attributed to Descartes - however what he actually said, is more like, "I am deceived, therefore I am" - In other words, in Descartes' search to find something about which he can be certain, he concludes that at least he is certain that he exists. Since even if he were deceived about everything else, there must be something which is being deceived, hence his own existence as a thinking thing (res cogitans) is impossible to doubt. So, just to make sure things are clear. After casting as much doubt  about all that he believes to be true, Descartes concludes that at least it is true that he exists.
That may not seem like a very great victory over absolute skepticism - and we will discuss in class what really can be deduced from such a certain axiom. For now, what is important is that you understand why Descartes believes that he has found something which is impossible to be false. Namely, the proposition, "I think, therefore I must be!"
 

he can never bring it about that I am nothing, so long as I shall be conscious that I am something

A related claim, "I do not think, therefore I am not" seems to follow deductively from the statement, "I think, therefore I am" This turns out to be true only if the original statement means, "If and only if I think, then do I exist". We examine this in detail in another class where symbolic logic is our main topic. In this introductory class, we shall visit this only in the form of the logical fallacy called denying the antecedent, and this fallacy will be central to understanding many logical/philosophical errors.  For now, I only want to point out that the related claim, "I do not think, therefore I do not exist" will be examined when we look at some of the implications of materialism (a problem Descartes will soon run into, as we shall see) and a subset of Philosophy called existentialism. In other words, we shall look at the claim that only if you can think do you exist. Hence, if you cannot think, (e.g. you are dead) then you do not exist.

I am, I exist, is necessarily true each time it is expressed by me, or conceived in my mind.

Here take note that the statement, "I am, I exist" is necessarily true each time it is expressed by me. This is different from saying that the statement taken alone, "I am, I exist" is true. Careful analysis of Descartes' reasoning explains why he makes this important qualification. Since the act of thinking ensures that the statement, "I am, I exist" is true, then this certainty only comes when someone is actually thinking it. Since thinking entails existence, thinking the thought, "I think, therefore I am" entails that the person thinking it indeed exists!

But I do not yet know with sufficient clearness what I am . . .

So now that Descartes has a starting point - a proposition that he knows for sure, it is now up to him to see what follows from that certain knowledge. Is he still in the same dilemma as before, knowing only one thing for certain and unsure about the rest, or can he deduce from this one certain piece of knowledge other things which can be known for certain?
 
 

I thought that I possessed a countenance, hands, arms, and all the fabric of members that appears in a corpse, . . .

Descartes avoids here some of the very questions that we shall tackle later on in this course, namely, what is meant or what follows from saying "I am a human being" - But back to Descartes point. Descartes here enumerates those things which he believes to be the most certain. You should take note that the elements he lists are directly involved in sensory experience. One's countenance (the face) is the seat of 4 of the body's major senses; sight, smell, hearing and taste. No wonder Descartes lists this body part first! Next comes primary elements related to the one remaining sense organ - touch.  Now a question. Formerly, Descartes listed as a proposition that was absolutely certain, the statement, "I am thinking, therefore  I am"- Could one not apply the same reasoning skills and conclude something like, "If I an seeing, therefore I am", or, "If I am feeling pain, therefore I am," or, "I'm am being tickled, therefore I am"? Actually, using the same logic, you can replace "I <insert any sensation that comes through the 5 senses>, therefore I am". The next question, is the act of thinking itself a sensation, or something different?
 

By body I understand all that can be terminated by a certain figure; that can be comprised in a certain place, and so fill a certain space as therefrom to exclude every other body; that can be perceived either by touch, sight, hearing, taste, or smell; that can be moved in different ways, not indeed of itself, but by something foreign to it by which it is touched [and from which it receives the impression];

This statement will later come back to haunt Descartes, by examining the very meaning of "body" or "corporeal" and what those words entail. To bring Descartes' definition of body to a more modern terminology - a body is something that can be enclosed (terminated by a certain figure), has a definite location in space, occupies a unique space which cannot be shared by other bodies (imagine trying to smash two coffee cups together so that they both occupy the exact same same location- you cannot do so, since they are both bodies and occupy unique locations which cannot be shared), and a body is something that can be detected by the senses.

likewise that of perceiving and thinking, I held as by no means pertaining to the nature of body

Descartes here answers the question I posed above, is "thinking" a sensation which properly belongs to the body, or something else? Or, better put, is "perceiving and thinking" part of the nature of a body the same way as just pure "seeing", "hearing", "smelling", "tasting" or "touching"? For Descartes, who is a dualist, the answer is  no. The mind (which comprises one's perceptions and thoughts") is distinct from the sensory organs of the body which bring about pure "seeing", "hearing", "smelling", "tasting" or "touching". So if we conclude, like Descartes, that the mind is distinct from the body - what else can we conclude?

But [as to myself, what can I now say that I am]

Descartes puts aside this diversion of thoughts and gets back to the main question. What if we indeed suppose that there is some powerful demon which is deceiving us in everything? After ascertaining that under this scenario, one can still conclude for certain that one exists whenever one is thinking, can one conclude anything else for certain?
 

any one of all those attributes

In other words, "seeing", "hearing", "smelling", "tasting" or "touching".

Thinking is another attribute of the soul; and here I discover what properly belongs to myself

Descartes rejects anything which has its source in the body. This  includes any physical activity, and any sensation or perception which is produced by the body - hence, "seeing", "hearing", "smelling", "tasting" or "touching" and physical activities can still be doubted and are not part of those things which make Descartes who he is. ONLY THOUGHT belongs to the soul or mind, and it is thought or the act of thinking which ensures one's existence. Therefore thought takes epistemic priority over sensation. The Cartesian dualism of mind [thought] and body [sensation] places thought on a higher and more essential ground.

The answer was, a thinking thing.

Not only can one be certain of one's existence, but one can be certain that one is a thing which thinks.

The question now arises, am I aught besides ?

Obviously it would be a small victory to conclude after such lengthy meditation that one is only a mind (something which is thinking) without a body. So in what follows, Descartes attempts to show by the same line of reasoning that he indeed must have a body.

Now it is plain I am not the assemblage of members called the human body; I am not a thin and penetrating air diffused through all these members, or wind, or flame, or vapor, or breath, or any of all the things I can imagine; for I supposed that all these were not . . .

To understand this passage, remember what Descartes has concluded. He has concluded that he is something that thinks. This he states he knows for sure. But this thing that thinks is not equal to the sum of his body parts, nor to some other "ghost-like" ephemeral substance. The mind [the thing that thinks] is distinct from the body. In other words, Descartes' metaphysical world consists of two types of "things" which are not the same, minds and bodies.
 

Is there also any one of these attributes that can be properly distinguished from my thought, or that can be said to be separate from myself ?

The passages leading up to this point are rather tedious. To summarize up to this point, Descartes is still debating what follows from his certain truth, "I think, therefore I am". He has distinguished thought (mind) from body - and ascertained that existence of the mind is certain. What, then about the body? Or perhaps better stated, what about those sensations (feelings) one thinks are produced by one's body (such as the pain one feels when you accidentally smack your finger with a hammer)?
The conclusion is that if one is certain about thinking then one can be certain about sensations as well. The "cause" of the sensation may be suspect - but the sensation itself is not. To use our mirage/lake example. While one may not be certain that there is a real lake which corresponds to one's visual sensation of the lake, one can indeed be certain that at least there is a visual sensation.

Visualize it this way:

Suppose you sense the sound of music.
One can suppose that this sound is falsely supplied to the mind by a powerful evil demon (sorry about the poor/cartoonish image, it was all I could find) - OR the sounds are coming to you from someone playing a musical instrument. Which supposition is correct? The evil powerful demon supposition or the real someone playing a musical instrument supposition?
Well, at this point Descartes is not ready to say. He does say, however, that the sensation of music is for certain, and it does not matter what the cause is as far as one's certainty about the sensation goes. While you may doubt that the music you are hearing is real or imaginary, or the product of some evil demon, IT IS IMPOSSIBLE TO DOUBT THE SENSATION OF MUSIC ITSELF!
 

EVIL DEMON (or some science fiction character) who is able to cause the mind to perceive whatever he wants, real or unreal. In this case he wants the "mind" (represented here as a pulsating material brain!) to perceive a sound.

MIND - senses the sound of music.

The world as we know it. In this case someone is really playing a musical instrument and the mind perceives the real sounds (as opposed to the unreal or "imaginary" sounds produced by Descartes' "evil demon")

I cannot help believing, that corporeal things, whose images are formed by thought [which fall under the senses] . . .

Corporeal things = material objects/bodies.
Remember Descartes knows at this point only the "sensations" of his mind. He is still unsure whether there are material objects which cause the sensations. But he confesses he seems to feel more certain about the existence of the material world than he does about his own mental world.
To understand this, think about using your eyes to examine a written page of any kind (both eyes and pages in books are members of the material world). If someone were to ask you to read the 5th sentence down from the top, 4th word over, any competent adult can do so.
Now take the book away and rely on your memories. Suppose someone were to ask you to examine your "mental image" of the written page and go down 17 lines and read the sixth word over! Most competent adults cannot do this. However, it is the realm of "mental images/sensations" which Descartes claims to know for certain, rather than the external material world.

What, then, was it I knew with so much distinctness in the piece of wax? Assuredly, it could be nothing of all that I observed by means of the senses, since all the things that fell under taste, smell, sight, touch, and hearing are changed, and yet the same wax remains.

Note that objects which belong to the material world are subject to change. In this wax example, Descartes notes that even though everything known about the wax by means of the sense (sight, smell, touch, taste, and feel) change, the wax remains the same. While this conclusion should seem odd (since this is the same Descartes who hypothesized about an evil demon who could deceive in all things) , nevertheless he maintains that something external to the senses must exist. But read carefully the next paragraph to see if you can detect the "faculty" which Descartes uses to come to this conclusion.

Let it be attentively considered, and, retrenching all that does not belong to the wax, let us see what remains. There certainly remains nothing, except something extended, flexible, and movable . . .

In other words, he wants to move from the particular example of wax, and see what remains of anything after you have subtracted all that is changeable from the object. What he concludes is that only the properties of extension, flexibility, and movability remain. Hence, any material object whatsoever is distinguished by these characteristics; extension, flexibility and movability. Since these are qualities of the material world, and distinguish material objects as such, they cannot be qualities or characteristics of the mental world.

I am, moreover, unable to compass this infinity by imagination, and consequently this conception which I have of the wax is not the product of the faculty of imagination.

This is an interesting sub argument. Let me illustrate it another way.
Imagine a piece of string, about 8 inches long. You can take this string, connect it end to end, and produce any number of different shapes with the closed loop of string. If I were to ask you, how many different closed shapes can you make with that loop of string - how would you answer? By counting each individual shape you can make (an when would you decide when you had made them all) - or by means of "intuition" you simply answer, "I can make countless shapes (or an infinity of shapes) with this string". Two different methods of coming up with an answer! The first involves tedious experiment and recording (counting the different shapes) - the other uses a faulty of the mind that Descartes here calls, "intuition" (we will discuss later if this is the best term to use) and comes up with an answer independent of any experiment which requires sensory input - namely that there are countless shapes one can make.
What then is intuition? Before, knowledge seemed to be linked to senses and combinations of sensory input. Here knowledge comes from something quite independent of sensory input. Is this something else about which one might be certain?

As an interesting side note, Jean Piaget, a Swiss experimental Psychologist, with a background in Biology and Philosophy, conducted tests on children to see if such concepts as "object permanence" were innate or learned. You can read about his interesting and often read essay, entitled "Genetic Epistemology", here!

I must, therefore, admit that I cannot even comprehend by imagination what the piece of wax is, and that it is the mind alone which perceives it. . .

In other words, "extension" is something that Descartes knows, but it is not something which can be perceived by means of the senses. Think of it this way, one can see colors, hears sounds, smell odors, feel sensations, but what does one perceive when they experience "extension" - This faculty of the mind, which is different from the sensory faculties might be called "reason" and,  in part, is one of the unique qualities possessed by humans and not animals.

and I am almost led into error by the terms of ordinary language . . .

The modern Philosopher, Ludwig Wittgenstein elaborated greatly on this point in his later years. For him, Philosophical problems resulted many times from the, "bewitchement of the intelligence by means of language." Descartes has a different end in mind, and indeed, one of Wittgenstein's main goals in his later years was to understand and unravel the philosophical problems which arose from the mind/body distinction found in  Discourse on Method.

But I judge that there are human beings from these appearances, and thus I comprehend, by the faculty of judgment alone which is in the mind, what I believed I saw with my eyes.

There is a distinction between the "appearance" of something (which comes through the senses) and what you judge it or interpret it to be (which is an act of the mind). An obvious example is when you see a round object from the side - the shape you actually see is more like an oval than a circle, but you judge the object to be round. This is an important distinction which we will discuss more in class.

What did I perceive which any animal might not have perceived ?

Descartes here compares sensory data (what one simply perceives) with the judgment faculty of the mind. For Descartes, the later is more certain, since after all, do not animals perceive and sense the world through sense organs the same as humans? Animals have no faculty of the mind for judgment or reason. The powers of the human mind to reason and interpret what it senses make the faculty of judgment more certain than the faculty of simple sense perception.

with how much greater distinctness must I now know myself, since all the reasons that contribute to the knowledge of the nature of wax, or of any body whatever, manifest still better the nature of my mind ?

Again, Descartes asserts that the "mind" is more knowable than things external to it (bodies - material objects). We shall see if he is able to go beyond this in his next Meditation.

and since they are not perceived because they are seen and touched, but only because they are understood . . .

Consider this point by way of this example. Suppose, for a moment, your philosophy instructor or someone else asked you to report something you know to exist. Following the example of Descartes, you look at a table next to you, and report that you at least know that the table exists as long as you are perceiving it. Descartes point is that the object you call table is made up of component "sensory perception" In other words, you see something that has a rectangular top, something else that looks skinny and long and are four in number, which you judge to be legs to the table - if you feel the table, your sensory organs report a certain smoothness of the surface - all of these sensations combined your mind - through a separate act - judges to be a table. This judgmental or interpretive aspect of mind must be distinguished from the simple act of raw experience or perception.