Summary
The discussion of substance and its properties is much more complex than the discussion of eternal truths, but once all the terms are sorted out it is not that difficult to grasp. There are three terms that are crucial to Descartes' understanding of basic ontology: substance, principal attribute, and mode. A substance is just a self-subsisting thing. A principal attribute is a special property of a substance, the property that makes that substance the kind of substance that it is. ("Principal attribute" is just another way of saying "essence".) A mode is any other property of a substance.
Descartes defines a substance as a thing that does not depend on anything else for its existence. That is to say, substance is a self-subsisting thing. Strictly speaking, then, only God is really a substance, because everything else depends on God for its existence (according to Descartes' picture, not only does God initially bring us into existence, but he must continually recreate us at every instant). More loosely speaking, though, anything that depends only on God for its existence counts as a substance.
The first thing to realize is that according to Descartes there are only three substances in the world: there is God, there is mind, and there is body. Jello is not one substance and gold another; both are the same substance of body. Since a principal attribute is the property that makes a substance the kind of substance that it is, there are also only three principal attributes: we have no idea what God's principal attribute is (if he even has one), but the principal attribute of mind is thought, and the principal attribute of body is extension.
The second important thing to realize is that the distinction between substance and principal attribute is only conceptual. That is to say, we have two concepts, (i.e. "principal attribute" and "substance") but these two concepts do not actually correspond to two different things out in the world. There is no such thing as a substance without its principal attribute. Body cannot exist without extension, and mind cannot exist without thought. This makes perfect sense when you consider that the principal attribute makes the substance what it is. How could a substance exist, Descartes asks, without being any particular kind of thing? It is impossible. In fact, not only can a substance not exist without its principal attribute (a metaphysical claim), a substance cannot even be clearly conceived without its principal attribute (an epistemological claim). How can you clearly conceive of something without clearly conceiving what sort of thing it is? The relationship between principal attribute and substance, then, is extraordinarily intimate.
The strong conceptual relationship between a substance and its principal attribute is what Descartes draws on in order to prove that the essence of mind is thought and the essence of body is extension. The proof of these claims rests on an unstated premise: P is the essence of S, if and only if I can conceive of S attributing only P to it. We can understand why Descartes would feel justified in asserting this premise given the strong conceptual relationship between a substance and its principal attribute. The principal attribute (or essence) is just that property that allows you to conceive of the substance. Whatever property that, all by itself, allows you to clearly conceive of the substance, then, must necessarily be the essence of that substance. (The full proof that thought is the essence of mind, and extension of body, will come in the next section.)
A mode, on the other hand, is much less intimately connected to a substance. A substance could exist without any particular one of its modes (though it could not exist without any modes at all). For instance, a substance could exist without being square, but it cannot exist without being shaped. A mode is actually just a particular way of being the principal attribute. A principal attribute is something determinable, (i.e. the property of extension or the property of thought) and a mode is a determinate way of being extended or thought (for instance, square is a way of being extended and imagining a unicorn is a way of being thought).
