What is combinatorics? I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is.

Could anyone give me a definition/explanation of combinatorics, of what combinatorics is, and what it deals with?
References that contain an answer to the question are appreciated.

Edit: I sees that many are saying that combinatorics deal with counting, but that doesn't seem to be the correct answer, for two reasons: first of all saying that combinatorics is just about counting means, at least to me, putting it inside set theory, because it's there where you define and deal with the more wide concept of counting; another reason is that there are some branch of mathematics which usually fall under combinatorics but doesn't directly deal with counting: for instance combinatorial design doesn't explicitly deal with counting.
 A: Personally, I see "combinatorics" as the "art of counting", which implies that the underlying objects are at least countable (= discrete), but better finite. I find it natural that "graph theory" is filed under "combinatorics" (because graphs are usually discrete, and there is a lot to count about graphs). 
The queen of combinatorics is generatingfunctionology.
A: 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.
A: My answer would be a branch of math that helps count things. 
Some of the simple counting questions are how many ways can you order things; How many ways are there to pick 12 donuts out of 3 styles at a donut shop; etc. Obviously, these are very simple counting questions and are usually covered in pre-combinatoric classes, but are part of Combinatorics.
I took a Combinatorics class last year and this was our book: Combinatorics - A Guided Tour. It was very helpful and easy to understand as long as you've had some higher math.
Hope this helps.
A: Have you looked at Section 1.1 ('How to count') in Stanley's book Enumerative Combinatorics Volume I? It doesn't give a concise answer to your question, but it does explain what combinatorialists do and thus answers your question to the extent that the answer is 'Combinatorics is what combinatorialists do'.
A: As a combinatorialist, I agree with the notion that combinatorics is roughly about "counting" things.  I disagree, however, with the assertion that such things must be discrete or even countable (countable here is meant in the mathematical sense).  To support this sentiment, I provide an example.
One of the ongoing projects which has emerged in my field is to count the number of $k$-point configurations in a particular space.  The well know Erdos-distance problem is one example:
Erdos (1946): Given a finite set $E \subset \mathbb{R}^n$, is it true that $E$ determines at least $|E|^{\frac{2}{n} + o(1)}$ distances?
There are also continuous versions of these problems.  For instance, let $C_k(E)$ denote the set of $k$-point configurations with vertices lying in $E \subset \mathbb{R}^n$.  What is the minimal value $\alpha > 0$ such that $\dim_{\mathcal H}(E) > \alpha$ implies that $C_k(E)$ has positive Lebesgue measure?  Here, $\dim_{\mathcal{H}} (E)$ denotes theHausdorff dimension of $E$.   Essentially, we want to "count" how many triangles we can find in a sparse (fractal) subset of a plane.  We do this by utilizing Hausdorff dimension.  In this sense, we are still "counting" objects, though the set of objects is uncountable.
A: According to Wikipedia:

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).

A: In The Two Cultures Of Mathematics, Tim Gowers offers a rather expansive concept:

I often use the word "combinatorics" not quite in its conventional sense, but as a general
  term to refer to problems that it is reasonable to attack more or less from first principles.
  (This is really a matter of degree rather than an absolute distinction.) Such problems
  need not be discrete in character or have much to do with counting. Nevertheless, there
  is a considerable overlap between this sort of mathematics and combinatorics as it is
  conventionally understood.

