Referring back to his logical work in the Categories, Aristotle opens book Zeta by asserting that substance is the primary category of being. Instead of considering what being is, we can consider what substance is.
Aristotle first rejects the idea that substance is the ultimate substrate of a thing, that which remains when all its accidental properties are stripped away. For example, a dog is more fundamental than the color brown or the property of hairiness that are associated with it. However, if we strip away all the properties that a dog possesses, we wind up with a substrate with no properties of its own. Since this substrate has no properties, we can say nothing about it, so this substrate cannot be substance.
Instead, Aristotle suggests that we consider substance as essence and concludes that substances are species. The essence of a thing is that which makes it that thing. For example, being rational is an essential property of being human, because a human without rationality ceases to be human, but being musical is not an essential property of being human, because a human without musical skill is still human. Individual people, or dogs, or tables, contain a mixture of essential and inessential properties. Species, on the other hand—for instance, people in general, dogs in general, or tables in general—contain only essential properties.
A substance can be given a definition that does not presuppose the existence of anything else. A snub, for example, is not a substance, because we would define a snub as “a concave nose,” so our definition of snub presupposes the existence of noses. A proper definition of a thing will list only its essential properties, and Aristotle asserts that only substances have essential properties or definitions. A snub nose, by contrast, has only accidental properties—properties like redness or largeness that may hold of some snubs but not of all—and per se properties—properties like concavity, which necessarily holds of all snubs but which is not essential.
Physical objects are composites of form and matter, and Aristotle identifies substance with form. The matter of an object is the stuff that makes it up, whereas the form is the shape that stuff takes. For example, the matter in a bronze sphere is the bronze itself, and the form is the spherical shape. Aristotle argues that form is primary because form is what gives each thing its distinctive nature.
Aristotle has argued that the definitions of substances cannot presuppose the existence of anything else, which raises the question of how there can be a definition that does not presuppose the existence of anything else. Presumably, a definition divides a whole into its constituent parts—for example, a human is defined as a rational animal—which suggests that a substance must in some way presuppose the existence of its constituent parts. Aristotle distinguishes between those cases where the parts of an object or definition are prior to the whole and those cases where the whole is prior to the parts. For example, we cannot understand the parts of a circle without first understanding the concept of circle as a whole; on the other hand, we cannot understand the whole of a syllable before we understand the letters that constitute its parts. Aristotle argues that, in the case of substance, the whole is prior to the parts. He has earlier associated substance with form and suggests that we cannot make sense of matter before we can conceive of its form. To say a substance can be divided by its definition is like saying a physical object can be divided into form and matter: this conceptual distinction is possible, but form and matter constitute an indivisible whole, and neither can exist without the other. Similarly, the parts of a definition of a substance are conceptually distinct, but they can only exist when they are joined in a substance.
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Sparknotes is a very useful resource, especially (so far) for Philosophy. I struggled with Plato's Republic and it really made it much clearer for me. Thank you!