Combinatorics is one of the most fascinating and frustrating branches of mathematics.For some bizarre reason no one really seems to understand,many mathematics students find it impossibly difficult while some find it as easy as breathing. It's also pretty difficult to precisely define.
Classically, combinatorics deals with finite sets of objects and the various ways their subsets and their elements can be counted and ordered.This definition seems the most reasonable to me.However, a number of mathematicians have vehemently disagreed. In fact,I once tried to define combinatorics in one sentence on Math Overflow this way and was vilified for omitting infinite combinatorics.I personally don't consider this kind of mathematics to be combinatorics, but set theory. It's a good illustration of what the problems attempting to define combinatorial analysis are.The best definition I can give you is that it is the branch of mathematics involving the counting and ordering of subsets of sets of objects.
As for textbooks, there's fortunately quite a few good ones. I first tried to learn combinatorics from the old classic Combinatorial Analysis by Liu. Much easier,well written and more informative is the terrific Introductory Combinatorics by Richard Brauldi. More modern and equally good are the books of Milkos Bona; A Walk Through Combinatorics and An Introduction To Enumerative Combinatorics Both books are outstanding, with the former being more comprehensive and the latter focusing more on counting techniques. Lastly, there's a terrific book by one of the great Hungarian masters; Discrete Mathematics by Laszlo Lovasz. Deep and complete, it's a really good introduction by one of the best practitioners.
That should get you started. One last thing: As I said, many talented mathematics students and mathematicians don't find this their cup of potion.As a result, you may find it frustrating and at times,begin to doubt your own mathematical ability. Keep in mind combinatorics has frustrated many an otherwise great mathematician and not to let it get to you. But it's hard to doubt that the skills learned in combinatorics are vital and important to the training of anyone interested in serious problem solving.