One way to think of categorification is that it's a generalization of enumerative combinatorics.  When a combinatorialist sees a complicated formula that turns out to be positive they think "aha! this must be counting the size of some set!" and when they see an equality of two different positive formulas they think "aha! there must be a bijection explaining this equality!"  This is a special case of categorification, because when you decategorify a set you just get a number and when you decategorify a bijection you just get an equality.  As a combinatorialist I'm sure you can come up with some examples that nicely illustrate how this sort of categorification is not totally well-defined. ("What exactly do Catalan numbers count?" hardly has a well-defined answer.)

A more sophisticated kind of categorification in combinatorics is ["Combinatorial Species"][1] which categorify power series with positive coefficients.

When people talk about categorification they usually mean something less combinatorial than the above two examples because they're almost always thinking of a different categorification of the natural numbers: Vector spaces.  Just like Sets vector spaces have a single invariant which is a nonnegative integer.  So when a combinatorialist sees positive numbers they think "aha! the size of a set" the typical categorifier (there are exceptions) thinks "aha! dimensions of vector spaces!"

Furthermore categorification is often dealing with things with more structure.  For example, if you're given a ring with a basis such that the product in that basis has positive structure constants (e.g. the Hecke algebra in the Kazhdan-Lusztig basis) you should think "this is Grothendieck group of some tensor category and the basis is the basis of irreducibles."  Similarly possibly negative integers can be thought of as dimensions of graded vector spaces.


  [1]: http://en.wikipedia.org/wiki/Combinatorial_species