except the clear and distinct perception of what I affirm . . .

The "clear and distinct" feeling one has when contemplating the knowledge of one's existence is the criteria Descartes will use later on to sort out what things are true, or beyond doubt, from things that are not certain. So, in other words, if one can perceive something clearly and distinctly with the mind, then that thing must be certain. Note that Descartes does not have sensual perceptions in mind (e.g. the visual image of something, or the auditory sense of a sound, or the feel of a touch) BUT ratherhe is talking about  knowledge that comes from deduction and analysis - the type of faculty of knowledge that is not part of the five senses, but is discovered by means of reason and logic.

They were the earth, the sky, the stars, and all the other objects which I was in the habit of perceiving by the senses. But what was it that I clearly [and distinctly] perceived in them ? Nothing more than that the ideas and the thoughts of those objects were presented to my mind. . .

In other words, Descartes doubts the existence of the external material world - but not internal perceptions of the material world. Perceptions, after all, are not doubtable as long as they are being perceived.

or make two and three more or less than five, in supposing which, and other like absurdities, I discover a manifest contradiction.

OK, let's see exactly what Descartes is talking about. Let suppose that 2+3 is equal to something other than 5, let's say 4 for example.
Now let's see if we can discover a contradiction!
First, remember what a contradiction is. The contradiction to the mathematical statement, "two plus three equal four", is "two plus three IS NOT equal to four".  Expressed mathematically it is written this way:
2+3=4
2+3 # 4 [where the "#" is read, "is not equal to" - (if anyone knows the code for the not equals sign in html, let me know!)]

So let's now assert 2+3=4 by making it a conclusion in an argument.

CONCLUSION: 2+3=4

Now we will assert some other statements which we will take as axiomatic premises  in our argument. I won't justify them here - that would require too much room. But I think they will be so obvious you will have no trouble

Let's define the symbols 2 through 5 as functions of the number 1.
Starting with the number 1 itself, the principle of identity (that everything is equal to itself) is stated as: 1=1
Making this a premise, we have:

PREMISE 1: 1=1

Now let's define the successor of 1 (the next number up) - which we will call "two" and symbolize it with "2". Obviously 2 cannot equal 1, since 1 is identical with 1 only, and 2, being different than 1, must be defined by another means. So for simplicity, we say 2=1+1. Hence:

PREMISE 2: 2=1+1

Now let's define the successor of 2 (the next number up) - which we will call "three" and symbolize it with "3". Obviously 3 cannot=1, nor can it equal 1+1, since 1 is identical with 1 only, and 2 is identical to 1+1;  SO 3 must be defined by 3=1+1+1! Hence:

PREMISE 3: 3=1+1+1

We continue in this same vein, and obtain premise 4 and 5:

PREMISE 4:  4=1+1+1+1
PREMISE 5:  5=1+1+1+1+1

NOW, what if I want to add 2+1? (notice, I am leaving the addition function undefined here, I take it  that it is obvious to the reader, a complete proof must define the addition term as well among other things)

Well, 2=1+1
and 1=1,
so 2+1 = 1+1+1,
BUT from looking at our premises we see that 1+1+1=3, hence 2+1=3!
 

Now, lets assemble all out our premises together.
PREMISE 1: 1=1
PREMISE 2: 2=1+1
PREMISE 3: 3=1+1+1
PREMISE 4:  4=1+1+1+1
PREMISE 5:  5=1+1+1+1+1

So now lets see if we can use our premises which are true BY DEFINITION (they are the axioms of our system).
Since we are interested in our what 2+3 equals, let's deduce it from our system.
From premise 2, we know 2=1+1
From premise 3, we know 3=1+1+1
Hence, 2+3 is the same as 1+1+1+1+1
By premise 5, 1+1+1+1+1=5!
SO we conclude that 2+3=5.

BUT WAIT, remember that we wanted to assert that 2+3=4, but we just proved that 2+3=5!
Hence, asserting 2+3=anything other than 5 produces a contradiction!! This is the type of manifest contradiction Descartes was referring to.

without the knowledge of these two truths, I do not see that I can ever be certain of anything . . .

If Descartes can prove there is a God, and likewise that the nature of that Deity is one that is not a Deceiver, then the slight metaphysical doubt about the rest of the external material world which was called into question by the evil demon hypothesis can be put aside.

but I also embrace in thought something more than the representation of the object; and of this class of thoughts some are called volitions or affections, and others judgments. . .

Descartes divides the mind into at different "faculties" - One faculty is merely the perception of ideas - "mental images" and the like. The other faculty has to do with "acts" of willing, judgment, emotions, and a third, which is involved in the act of reasoning. Notice that at this point, only these ideas, wills, judgments, emotions, and reasoning are known. Whether or not material bodies which exist outside the mind and correspond to the ideas or mental images which are in his mind is yet to be established.

Now, with respect to ideas, if these are considered only in themselves, and are not referred to any object beyond them, they cannot, properly speaking, be false; for, whether I imagine a goat or chimera, it is not less true that I imagine the one than the other.

Again Descartes affirms that one can be certain of one's thoughts, but cannot be certain as to whether anything corresponds to those thoughts in the real world. In other words, if you "imagine" some monster or creature in your head, there can be no doubt about the mental image - there is considerable doubt, however, as to whether your imaginary creature exists in the real world outside of your head!
 

hippogryphs
An imaginary creature, half dragon, half horse. From the Greek words, hippos (horse) and probably gryps [psi rather than phi] (griffin or dragon) - Liddel and Scott, An Intermediate Greek English Lexicon, Oxford University Press, Oxford: 1991.
 

What I have here principally to do is to consider, with reference to those that appear to come from certain objects without me, what grounds there are for thinking them like these objects.

Here and in next passages that follow, Descartes tries to show that the objects in the material world which "seem" to correspond to his mental ideas about such objects indeed correspond in the way one naturally thinks about them. He gives two possible reasons; one, that he is taught so by nature, and two, that those ideas are not dependent on his will. Let's examine each of these in turn.

"That he is taught so by nature" - As you will see in paragraph nine, he means by this a sort of intuition that leads him to believe that ideas correspond to material objects (remember an "idea" is like a mental image - and as we all know, mental images need not correspond to anything in the real world). This intuition, or "spontaneous impetus" seems something like "habit" which itself offers no assurance of being true.

"Ideas are not dependent on my will" - THis is the major reason Descartes believes in a  "resemblance between ideas and their objects" - since most of the time, one's mental images and sensations about the world come about forcibly to the mind. Compare closing your eyes and imagining the room you are in- versus seeing the room with your eyes open. In the "closed eyes scenario" one's ideas of the room are  "dimmer" and less detailed, whereas with one's eyes open, the room presents itself as it is, and it is not subject to change do to the whim on one's imagination or inability to recall detail.
 

And, in fine, although I should grant that they proceeded from those objects, it is not a necessary consequence that they must be like them. . .

So, even if we grant that one's ideas correspond to something in the material world, it does not follow that those ideas resemble the things in the external material world they represent.

I find in my mind two wholly diverse ideas of the sun;

One's visual sensation of the sun tells the person that the sun is small in size. This knowledge comes about directly through a sensation (visual) of the sun - however, via other a logical method of reasoning, one can certainly conclude that the sun is much larger than it appears - and it is this faculty of reasoning which takes epistemic priority over even pure "visual" images.

On a side note: The Greek Philosopher and Astronomer Hipparchus of Rhodes discovered using logic and trigonometry that the Sun was far more distant than the moon - click here for his proof.

but, considering them as images, of which one represents one thing and another a different, it is evident that a great diversity obtains among them . . .

So Descartes compares his ideas, to see which ones have greater epistemic power. First, lets summarize the division of ideas. Remember first that an idea simply is a representation in the mind. Hence we have:

1. Ideas which seem to represent things in the material world (e.g. the what you just simply see, hear, feel, touch or taste when you look at the world around you)

2. Ideas which come from some other faculty which is not itself one of the five senses, innate, adventitious, and factitious. In case you did not follow the link in the text that provided a definition of these words, I will provide it here:

Innate -Originating in, or derived from, the constitution of the intellect, as opposed to acquired from experience;

Adventitious - Added extrinsically; not essentially inherent; accidental or causal; additional; supervenient; foreign.

Factitious- Made by art, in distinction from what is produced by nature; artificial; sham; formed
by, or adapted to, an artificial or conventional, in distinction from a natural, standard or rule; not natural; as, factitious cinnabar or jewels; a factitious taste.

3. Ideas which come from a faculty of reason and logic (e.g. that the sun is larger than its visual appearance)

Comparing the epistemic quality of all these types of ideas, what proceeds from reason and logic has the greatest epistemic value.
THe idea that there exists a Deity, all knowing, all powerful, all present and all benevolent is supposedly one of the types of knowledge which comes from a faculty of reason - and known more than any other type of idea.
 

Now, it is manifest by the natural light that there must at least be as much reality in the efficient and total cause as in its effect; for whence can the effect draw its reality if not from its cause ?

In what follows, Descartes offers a type of "proof" of God's existence. We will examine this proof later on in the course when we look at on of Snt. Thomas Aquinas' (1255-1274) ways for proving the existence of God.

To greatly simply things for now. Descartes is saying that in the relationship of "cause and effect",  no effect can have more than what was in the cause. This was paraphrased by Snt. Thomas Aquinas as, "ex nihilo, nihil fit" or from nothing, nothing comes to be. In our popular culture we have the saying, "you can't get nothing from nothing"  Keep this in mind as you read the next paragraphs, and that will help you understand what Descartes is saying. And here it is important to note that Descartes wants to prove the existence of God (NOT the evil-demon) and he is using a faculty of reason, which has greater epistemic value than the senses or imagination, to do so.
 
 

And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality

Here Descartes' knowledge of Snt. Thomas Aquinas, and  Snt. Thomas' arguments for the existence of God which Descartes was exposed to during his Jesuit education, comes out in the Meditations. We will examine these arguments in detail soon. This particular argument is also known as the argument from first cause (the unmoved mover) - and represents another step in Descartes' reasoning to establish the existence of God.

It is this: if the objective reality [or perfection] of any one of my ideas be such as clearly to convince me, that this same reality exists in me neither formally nor eminently, and if, as follows from this, I myself cannot be the cause of it, it is a necessary consequence that I am not alone in the world, but that there is besides myself some other being who exists as the cause of that idea; while, on the contrary, if no such idea be found in my mind, I shall have no sufficient ground of assurance of the existence of any other being besides myself, for, after a most careful search, I have, up to this moment, been unable to discover any other ground.

In other words, Descartes asks the question as to what could be the cause of the most certain of all his ideas.
You should recall the ideas which come about by means of the faculty of reasoning and logic are the most certain.
Since, following his former line of argument, the cause of something must be greater than its effect.
That being the case,  what could be the cause of Descartes reasoning abilities (the effect)?
Since effects cannot be greater than causes, that rules out Descartes himself.
In other words, Descartes' ability to reason must proceed from something greater than he.

THIS is not the case when one talks about angels, or other animals, other people, and the like. Since these ideas, much like the idea of a dragon or some other imaginary creature, can simply be made up hodge-podge from other ideas, such that their cause does not exceed their effect (in this case the cause of the idea of a hippogryph, for instance, is simply the sum of the effects of other material objects combined in unique ways in his mind).
 

As belonging to the class of things that are clearly apprehended, I recognize the following, viz, magnitude or extension in length, breadth, and depth; figure, which results from the termination of extension; situation, which bodies of diverse figures preserve with reference to each other; and motion or the change of situation; to which may be added substance, duration, and number. . .

Here Descartes lists those things which are certain (clearly and distinctly perceived) about material bodies. In more modern terms, they are number, extension in space (length, width, height), they are bounded (meaning objects have a finite size, where one can determine an end of one object and the beginning of something else), a specific location in space, motion (the ability to detect change in spatial location), duration in time, and quantity (singular, plural, etc.)
 
 
 

By the name God, I understand a substance infinite, [eternal, immutable], independent, all-knowing, all-powerful, and by which I myself, and every other thing that exists, if any such there be, were created. But these properties are so great and excellent, that the more attentively I consider them the less I feel persuaded that the idea I have of them owes its origin to myself alone. And thus it is absolutely necessary to conclude, from all that I have before said, that God exists.

Now Descartes turns his attention to what is meant by God. After defining God, he then examines whether any of the adjectives that describe God could originate in himself. He concludes that they can not. Hence, from the premise that causes cannot produce effects greater than themselves, and the additional premise that Descartes cannot himself produce the ideas he has about God, then it follows that God must be the ultimate cause of these ideas - hence God exists.
 
 

But perhaps I am something more than I suppose myself to be, and it may be that all those perfections which I attribute to God, in some way exist potentially in me, although they do not yet show themselves, and are not reduced to act. . .

The most obvious counter argument to the above would be that Descartes himself is a God, or at least has the potential to be a God. Then the cause of Descartes' ideas about God would be in himself and not exceed the effect, since they would be one and the same.

still all these excellences make not the slightest approach to the idea I have of the Deity, in whom there is no perfection merely potentially [but all actually] existent;

Descartes discounts the above counter-argument by claiming that there is a distinction in being a potential God, versus being an actual God, and since Descartes clearly and distinctly has in mind an actual God, then the ideas about that actual Deity cannot rest in Descartes alone.