What precisely Is "Categorification"?

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?" has many answers rather than a single right answer.)

A more sophisticated kind of categorification in combinatorics is "Combinatorial Species" 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.

The Wikipedia answer is one answer that is commonly used: replace sets with categories, replace functions with functors, and replace identities among functions with natural transformations (or isomorphisms) among functors. One hopes for newer deeper results along the way.

In the case of work of Lauda and Khovanov, they often start with an algebra (for example ${\bf C}[x]$ with operators $d (x^n)= n x^{n-1}$ and $x \cdot x^n = x^{n+1}$ subject to the relation $d \circ x = x \circ d +1$) and replace this with a category of projective $R$-modules and functors defined thereupon in such a way that the associated Grothendieck group is isomorphic to the original algebra.

Khovanov's categorification of the Jones polynomial can be thought of in a different way even though, from his point of view, there is a central motivating idea between this paragraph and the preceding one. The Khovanov homology of a knot constructs from the set of $2^n$ Kauffman bracket smoothing of the diagram ($n$ is the crossing number) a homology theory whose graded Euler characteristic is the Jones polynomial. In this case, we can think of taking a polynomial formula and replacing it with a formula that inter-relates certain homology groups.

Crane's original motivation was to define a Hopf category (which he did) as a generalization of a Hopf algebra in order to use this to define invariants of $4$-dimensional manifolds. The story gets a little complicated here, but goes roughly like this. Frobenius algebras give invariants of surfaces via TQFTs. More precisely, a TQFT on the $(1+1)$ cobordism category (e.g. three circles connected by a pair of pants) gives a Frobenius algebra. Hopf algebras give invariants of 3-manifolds. What algebraic structure gives rise to a $4$-dimensional manifold invariant, or a $4$-dimensional TQFT? Crane showed that a Hopf category was the underlying structure.

So a goal from Crane's point of view, would be to construct interesting examples of Hopf categories. Similarly, in my question below, a goal is to give interesting examples of braided monoidal 2-categories with duals.

In the last sense of categorification, we start from a category in which certain equalities hold. For example, a braided monoidal category has a set of axioms that mimic the braid relations. Then we replace those equalities by $2$-morphisms that are isomorphisms and that satisfy certain coherence conditions. The resulting $2$-category may be structurally similar to another known entity. In this case, $2$-functors (objects to objects, morphisms to morphisms, and $2$-morphisms to $2$-morphisms in which equalities are preserved) can be shown to give invariants.

The most important categorifications in terms of applications to date are (in my own opinion) the Khovanov homology, Oszvath-Szabo's invariants of knots, and Crane's original insight. The former two items are important since they are giving new and interesting results.

I think one important point that has been missed here is that there is not (currently) a precise answer to this question.

There is a loose answer along the lines of that which Pete Clark gave, but I think there may be a typo in that response. And, of course, there are specific instances which shed light (and provide new mathematics) as Scott has pointed out.

"As you can see from this article, one aspect of categorification is the systematic negation of "decategorification", which is the process of taking a category (e.g. the category of sets) and replacing it by the class of isomorphism classes of its objects (in that case, the class of cardinal numbers)."

Categorification is NOT the systematic negation of decategorification. Decategorification can be defined in various ways as a systematic process and categorification can be understood as the non-systematic (i.e. creative) process of undoing decategorification.