In general, the center (or centre) of an algebraic object $X$ is the collection of things which “commute with all elements of $X$.” This has a number of specific incarnations.
The original example is the center $Z(G)$ of a group $G$, which is defined to be the subgroup consisting of all elements $g\in G$ such that for all elements $h\in H$ the equality $g h=h g$ holds. The center is an abelian subgroup, but not every abelian subgroup is in the center. See also centralizer.
This notion of center of a group can be generalized to the center of a monoid in an obvious way.
The center of a Lie algebra $L$ is an abelian Lie subalgebra $Z(L)$, consisting of all elements $z\in L$ such that $[l,z]=0$ for all $z\in L$. There are generalizations for some other kinds of algebras.
The center of a monoid can be horizontally categorified to the center of a category. Specifically, the center of a category $C$ is defined to be the commutative monoid $[C,C](Id_C,Id_C)$ of endo-natural-transformations of the identity functor of $C$. It is straightforward to check that this reduces to the usual definition if $C = \mathbf{B}(A,\times)$ is the delooping of a monoid.
For a generator $G$ of a category $\mathcal{C}$ there is an embedding of $Z(\mathcal{C})$ into the monoid $Hom(G,G)$ given by $\eta\mapsto\eta _G$. In particular, if $Hom(G,G)$ or $Z(Hom(G,G))$ is trivial, as happens e.g. for $Set$ with $G=\ast$, then so is $Z(\mathcal{C})$ (Hofmann 1975).
For Cauchy complete $\mathcal{C}$ the idempotent elements of $Z(\mathcal{C})$ correspond precisely to the quintessential localizations of $\mathcal{C}$ (Johnstone 1996).
The notion of center can also be vertically categorified. It is easy to categorify the notion of center of a category as defined above: if $C$ is an n-category, then its center is the monoidal $(n-1)$-category $[C,C](Id_C,Id_C)$ of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of braided monoidal $(n-1)$-category, just as the center of a category is actually a commutative monoid, not merely a monoid.
For instance if $C = \mathbf{B}_\otimes \mathcal{C}$ is the delooping of a monoidal category, then this center is called the Drinfeld center of $(C, \otimes)$.
Generally, we can now obtain a notion of the center of a monoidal $n$-category by regarding it as a one-object $(n+1)$-category, according to the delooping hypothesis. It follows that the center of a monoidal $n$-category should naturally be a braided monoidal $n$-category. This is known to be true when $n=0$ (the center of a monoid is a commutative monoid) and also for $n=1$ and $n=2$.
Note that a monoidal $n$-category has two different centers: if we regard it as a one-object $(n+1)$-category, then its center is a braided monoidal $n$-category, but if we regard it merely as an $n$-category, then its center is a braided monoidal $(n-1)$-category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal $n$-category has three different centers, depending on whether we regard it as an $n$-category, a connected $(n+1)$-category, or a 2-connected $(n+2)$-category, and so on (a $k$-tuply monoidal $n$-category has $k+1$ different centers).
It seems that in applications, however, one is usually most interested in the sort of center of a monoidal $n$-category $C$ obtained by regarding it as a one-object $(n+1)$-category, thereby obtaining a braided monoidal $n$-category. It is in this case, and seemingly this case only, that the center comes with a natural forgetful functor to $C$, corresponding to the classical inclusion of the center of a monoid. (For $n\gt 0$, however, this functor will not be an inclusion; the objects of the center of $C$ are objects of $C$ equipped with additional structure.)
Moreover, one expects that if we perform this “canonical” operation on a k-tuply monoidal n-category (for $k\ge 1$), the resulting braided monoidal $n$-category will actually be $(k+1)$-tuply monoidal. This is known to be true in the cases $n\le 4$: the center of a braided monoidal category is symmetric monoidal, the center of a braided monoidal 2-category is sylleptic, and the center of a sylleptic monoidal 2-category is symmetric.
Finally, if we decategorify further, we find that the center of a set (i.e. a 0-category) is a monoidal (-1)-category, i.e. the truth value “true.” This is what we ought to expect, since when $C$ is a set, there is precisely one endo-transformation of its identity endofunction (namely, the identity).
An old query about the categorical notion of center is archived at $n$Forum here.
A special case is the center of an abelian category which has a special entry because of a number of special applications and properties.
See center of an ∞-group.
R.-E. Hoffmann, Über das Zentrum eine Kategorie , Math. Nachr. 68 (1975) pp.299-306.
P. Johnstone, Remarks on Quintessential and Persistent Localizations , TAC 2 no.8 (1996) pp.90-99. (pdf)