Categorification can be thought of as the process of replacing equalities with isomorphisms (in some category). A basic example is replacing a numerical combinatorial identity such as
$$2^n = \sum_{k=0}^n {n \choose k}$$
with a bijection between two sets (an isomorphism in $\text{Set}$); in this case, between the set of subsets of $\{ 1, 2, ... n \}$ and the disjoint union of the sets of subsets of size $k$ over all $k$.
As another example, the category $\text{FinSet}$ of finite sets itself categorifies the rng $\mathbb{N}_{\ge 0}$ of non-negative integers; the coproduct categorifies addition and the product categorifies multiplication. (Also taking Hom-sets categorifies exponentiation.) In this way we replace equalities such as $1 + 1 = 2$ with isomorphisms $1 \sqcup 1 \cong 2$ (where $1$ is the set with one element, $2$ is the set with two elements, and $\sqcup$ is the coproduct).
Schur functors categorify the theory of symmetric functions (specifically of Schur functions). Taking the direct sum of Schur functors categorifies addition of symmetric functions, taking the tensor product categorifies multiplication of symmetric functions, and composing categorifies plethysm. So here we replace equalities between expressions involving symmetric functions with natural isomorphisms between expressions involving Schur functors.
As far as the linked arXiv paper goes, the idea is to replace a ring $A$ by a category $C$ with direct sums and tensor products such that the latter distributes over the former, such as the category of $(R, R)$-bimodules for some ring $R$. Equality of elements in $A$ is then replaced by isomorphism between objects in $C$. Taking the Grothendieck group of $C$ (a form of decategorification) should then recover the original ring.
One reason to attempt to do this is that the category $C$ may have a distinguished set of objects that provides a distinguished set of generators for $A$. For example, the Hecke algebra $H_n$ of $S_n$ turns out to be the Grothendieck group of the category of Soergel bimodules. Among these we can distinguish the indecomposable modules, and these turn out to give the Kazhdan-Lusztig basis of $H_n$.
