How cool is it that a category object in the category of groups is the same thing as a group object in the category of categories (or even in category of groupoids)? There's just no obvious reason why swapping the words "groups" and "categories" should lead to equivalent definitions. I guess this is one of those cases in which category theory validating all the things you would want to be true.
I've been reading this short expository article, which explains the relationships between these and crossed modules, all of which are apparently incarnations of (strict?) 2-groups. I don't understand crossed modules at all yet, but I'm looking forward to working out for myself what the smallest finite strict 2-groups are, concretely. Maybe later this year I'll be able to read Baez & Lauda's notes on 2-groups.
It's crazy that category theorists use the term "2-group", though, since that has long meant something totally different in group theory: a 2-group in the original sense is a group whose elements all have order a power of 2, which is a completely unrelated concept.