Descartes' discussion of essence is intended as a strong reaction against Aristotelian empiricism. According to Aristotle, we learn the essence of, say, a triangle, by examining instances of triangular-shaped objects in the world and extracting the essence of triangles from these worldly instances. Descartes turns this formulation on its head, saying that we learn the essence of a triangle solely through the intellect, and only afterward do we look at the real world and see if there are instances of triangles. In Descartes' formulation, whatever properties of a thing that we clearly and distinctly perceive must be essential. Thus, bodies are essentially extended, since extension is clearly and distinctly perceived by the intellect prior to any empirical investigation. The essence of body will be further discussed in the Sixth Meditation.
Descartes provides strong reasons to support his move against Aristotle. If essences are extracted from the real world, how is it that we understand perfect triangles when there are none? And more importantly, how do we understand abstract shapes that we have derived mathematically but have never encountered in the real world? In Aristotle's defense, however, we could point out that while Descartes has given us a better understanding of the essences of mathematical objects, he has left us completely in the dark as to how we can know the essence of material things. Is it possible to apply the intellect to understanding the essence of, say, gold, without ever encountering gold in the world?
Descartes then turns his discussion of essence toward a second proof of God's existence. This proof is weaker than the one found in the Third Meditation, and we might wonder why he adds it. Is he not certain that his earlier proof is satisfactory? And if so, what holes does this new proof patch up? Most importantly, it reinforces the connection between God and clear and distinct perceptions. Clear and distinct perceptions are made certain because God exists, and God's existence, as an essential property, is clearly and distinctly perceived. Of course, this reinforced connection only reinforces the conundrum of the Cartesian Circle. This problem is discussed in the commentary to the Third Meditation, Part 3.
The proof of God's existence found here is a version of a proof that was popular among the Scholastic philosophers. Our idea of God is the idea of a perfect being, and one of the attributes of a perfect being would be existence, since it is more perfect to exist than not to exist. In Descartes' formulation, existence is not just an attribute, but an essential property of God's, so that God cannot be conceived of without existence. This proof, however, rests on the faulty assumption, first pointed out by Kant, that existence is a predicate or a property, like "being red" or "being tall." In fact, "exists" is a very different kind of predicate than "is red" or "is tall." The predicate "exists" does not modify an object so much as it modifies the world. If I say "the red car exists," the property of redness is something that modifies the car. On the other hand, "exists" does not modify the car so much as it says that the world is such that the car is in it. In that sense, "exists" is not a property of the car's.