TITLE: What is a "category"?
QUESTION [9 upvotes]: I have tried to understand category theory for a while but never been able to get it. I finally found a text I like called Category Theory for Scientists. However, I think it would be easier to read this text if I knew roughly what category theory is about. Since I'm new to SE, I thought I'd come here and see if I can get some help before trying again. 
Can someone summarize a few of the most important, elementary concepts in category theory? For example, what is a "category"? 
Thanks!

REPLY [2 votes]: In math we often have objects and special functions between them.  For example:

Sets and all functions
Vector spaces and linear transformations
Groups and homomorphisms
Topological spaces and continuous maps

That's basically all a category is.  You have objects and you have morphisms (the functions).