asked the question: how can we know
anything? Doesn't all our knowledge
depend on other knowledge? How exactly? Is there anything we can be absolutely sure of, without depending on any assumptions?
And his answer: regardless of anything else, what I cannot doubt is that I'm pondering this question! Cogito, ergo sum.
From this starting point, Descartes proceeded to 'prove' the existence of many other things; God among them.
I agree with Little Tony that this whole enterprise is a gross overestimation of the powers of introspection and logical reasoning. Descartes puts human reason at the center of the universe. While he was not the first or last person to do so, he was very influential. Maybe, just maybe, thought isn't the start and end of everything.
And if you hate Descartes for making people believe that it is, taking him to court won't help: if any place is built on Cartesian thinking it's the courtroom. Come to think of it, there's no such thing as truth anyway, all we do in courtrooms is make up excuses. Don't bother with all that crap, just take the guy out and shoot him.
OK, calm down, calm down. Logical argument isn't all-powerful but it does have its place. Descartes truly wasn't interested in his ego, in putting his own mind above anyone else's, but in the opposite: he was looking for objective communicable truth, knowledge we can all share about the world. While it was a little one-sided of him to rely on logical argument only, this attitude has led him and others to many powerful insights about the world.
In particular, it leads to mathematics. To take his famous phrase in context, you have to know that Descartes was a mathematician; his interest was in the power of deductive reasoning, as practiced in Euclidean geometry. Descartes was an excellent mathematician: we all learn some of his work in school, and it has many important practical applications, mostly in physics. He may have overplayed his hand with the cogito, ergo sum business, but that doesn't take away any of the merits of his approach in other areas.