A category is some kind of generalization of a mathematical theory. An example of a mathematical theory is the theory of groups. Important concepts in the theory of groups are the groups itself and the homomorphisms between the groups. These two concepts give rise to the category of groups.
Another example of a mathematical theory is topology. Important concepts in topology are the topological spaces and continuous maps between them. These two concepts give rise to the category of topological spaces.
In general, every good mathematical theory should have a set of objects and morphisms between those objects. It are these concepts which define a category.
I added the notion of a functor. I'll expand it in the following days.
Also, the notion of "locally small category" is not standard at all. Of the three main books about categories (Borceux, Maclane, and Adamek-Herrlich-Strecker), only Maclane talks about locally small categories. Furthermore, I've not encountered the notion of locally small categories very much in research papers. But there might be some who do use it...
Hi micromass! I think the reader will wonder what it's all for… is it just an interesting definition? Perhaps some useful applications should be added, eg functors and a theorem or two? (A separate entry "functor" would be pointless here, it would get no autolinking. )