Any unit of discourse, oral or written, in which reasons are given in support of a claim, is an argument. The claim is called the conclusion and the reasons offered in support are called the premises. This is a very general definition and applies to many different kinds of logic. When we look at categorical logic, we can be more precise about its rules but we may need a few more definitions. The simplest form of a categorical argument is know as the categorical syllogism - three categorical statements relating three and only three classes of things.
All crows are birds.
Freddie is a crow. Freddie is a bird. The general pattern that is shown here is common to all categorical syllogisms. Once you understand that concept, you understand most of categorical logic. It's really not very complicated and it doesn't really help very much when you're dealing with complex problems, but it took humans a very long time just to get this far and thus it seems reasonable to start with this class of logic as a jump-off point.