A category is a collection of objects with a notion of morphism between said objects. Morphisms are really not defined much beyond that; it is all left rather a

A category is a collection of objects with a notion of morphism between said objects. Morphisms are really not defined much beyond that; it is all left rather abstract, though there are some basic laws that must be satisfied. Categories are relatively uninteresting without functors. Functors are more or less mappings or transformations between categories. Category theory by some accounts initiated in the foundations of algebraic topology. In that subject, one wishes to codify some of the aspects of a high-dimensional shape or geometric object in algebraic terms. The hope is that the algebraic image of the shape is easier to understand than the shape itself. The mapping of shapes into their algebraic "shadows" is a functor. For instance, you can take a torus and look at the collection of all closed loops drawn on the surface of the torus. Now identify two loops as the same if one can be smoothly deformed into the other by moving it along the surface of the torus. Long story short, this gives the set of such loops a structure known as a group. There is a very extensive theory developed around groups, so you might surmise that you could partially classify shapes by looking at this associated group (called the fundamental group or first homotopy group). This mapping from geometric objects to groups of loops drawn on them is a functor from the category of topological spaces to the category of groups. This is the beginning of homotopy theory. As for your comment on sets, there is the category Set which consists of all sets. That is, the objects of Set are sets and, given two sets X and Y, the morphisms between X and Y in Set are simply the functions f : X -> Y. An interesting technical point here (which is related to your quote) is that Set itself cannot be a set, as there is no set of all sets within Zermelo-Fraenkel set theory (the most common axiomatic formulation of set theory). Rather, there is a notion of class, and one says that the collection of all sets forms a class rather than a set.