Categorification, is the act of viewing an algebraic object as the "shadow" of some category. So the category $\mathcal{C}$ is a categorification of $A$ if $A$ can be viewed as an algebraic invariant of $\mathcal{C}$. This is a loosely defined concept, so it's easier to explain with a list of examples.

Examples

• The category $\text{Vect}_\mathbf{k}$ of vector spaces over a field $\mathbf{k}$ can be seen as a categorification of the integers via the Grothendieck group construction. The Grothendieck group $K_0(\text{Vect}_\mathbf{k})$ is isomorphic to the integers.
• The construction of the Hall algebra of a category is another source of categorifications. In particular, if $\mathfrak{n}^+$ is the upper half of a Lie algebra $\mathfrak{g}$, then the quantized enveloping algebra $U_q(\mathfrak{n^+}$ is isomorphic to the Hall algebra of the category of representations of a quiver based on the Dynkin diagram of $\mathfrak{g}$. In this sense, that category of representations categorifies the quantized enveloping algebra.
• What precisely Is “Categorification”? on MathOverflow
• Examples of Categorification on MathOverflow