Part Four of the Discourse reads as a very brief summary of the first three Meditations (though the geometrical proof of God's existence is in the Fifth Meditation). A more detailed commentary on all these matters can be found in the SparkNote on the Meditations. This commentary will simply be a brief overview.
At the beginning of his investigation, Descartes undertakes to consider as false everything that he can possibly doubt. Such doubt effectively demolishes the whole enterprise of Aristotelian philosophy, which bases its claims on sensory experience and demonstrative reasoning. His goal is to sweep away the philosophical prejudices of the previous two thousand years and to start afresh. In doing so, he also manages to set the tone for the nearly four hundred years of philosophy that follow him. The questions of how we can know that there are objects external to our minds, that there are minds other than our own, and so on, have been hotly contested in the light of Descartes's new standard for what counts as certainty.
Perhaps Descartes's most significant contribution to philosophy is his revolutionary conception of what the human mind is. According to Aristotelian philosophy, only reason and understanding are distinctly mental properties. Sensing, imagination, and willing are not simply mental properties, since they connect the mind with objects in the world. Descartes overturns this conception, suggesting that our sense experience, imagination, and will are all a part of the mind alone, and are not linked to the world. In suggesting that we may be dreaming or otherwise deceived, Descartes argues that sensory experience is not necessarily a faithful report of what is actually in the world. Effectively, Descartes re-conceives the mind as a thing—the source of all the thoughts, sensations, imaginings, and so on that constitute our world—trapped inside our body. How our mind can connect with a world outside this body has been a pressing problem for all modern humans since Hamlet.
"I am thinking, therefore I exist" is Descartes's proposed way out. This famous phrase is less precisely translated as "I think, therefore I am." The fact that I am thinking right now, and not that I am capable of thought, is what confirms that I exist right now, and not that "I am" in general. Descartes cannot doubt that he exists, and so he claims to have certain knowledge of this fact. It is quite tricky, however, to determine the nature of this knowledge. Descartes has doubted the certainty of demonstrative reasoning, so it can't follow from a logical argument. Descartes's answer is that it is a "clear and distinct perception": it is not something he has to argue for; it is something that it is simply impossible to doubt.
Descartes seems to argue in a circle later in his discussion, when he claims that God confirms the truth of clear and distinct perceptions. This implies that without God, clear and distinct perceptions would not be true. But he has only managed to "prove" that God exists by appealing to a clear and distinct perception to that effect. What, then, is the foundation upon which Descartes builds? If God is the source of all truth, including the truth of clear and distinct perceptions, how can Descartes prove that God exists? And if clear and distinct perceptions are the source of all truth, then what role does God play in all this?
We should note that Descartes's "proofs" of God are neither original nor very satisfying. Unlike his revolutionary ideas about the nature of the mind and of certainty, his proofs of God are borrowed from the medieval scholastic tradition. The first proof claims that the idea of God, as an idea of perfection, must be caused by something as perfect as the idea itself. This proof relies on notions of causation that are questionable to say the least. The second proof claims that existence is a property of God just as geometrical figures have certain properties. Kant was the first to point out that "exists" is not a property in the way that "angles add up to 180 degrees" is. Having angles that add up to 180 degrees is a property of a triangle: it says something about the triangle. Existing, however, is not a property of God's so much as it is a property of the world: it is saying that the world is such that God exists in (or above) it.