Here are some notions of "finiteness" which make sense in lots of categories, and which characterize the finite sets when the category is $Set$. I view this as an addendum to the notions discussed in Hanul Jeon's answer. A salient point is that notions of finiteness which agree in $Set$ may diverge when considering more general categories, even when those more general categories are things like toposes, i.e. "generalized categories of sets".

A set $X$ is finite if and only if it is Noetherian, i.e. it satisfies the ascending chain condition for subobjects.

A set $X$ is finite if and only if it is Artinian, i.e. it satisfies the descending chain condition for subobjects.

A set $X$ is finite if and only if it is Hopfian, i.e. every surjection $X \to X$ is a bijection.

A set $X$ is finite if and only if it is co-Hopfian, i.e. every injection $X \to X$ is a bijection.

In the 1-topos literature, the term "Dedekind-finite" is used to mean co-Hopfian.

- A set $X$ is finite if and only if it is finitely-presentable, i.e. for every chain $Y_0 \to Y_1 \to \dots$ and every function $f: X \to \varinjlim Y_n$, there is some $m$ such that $f$ factors through $Y_m \to \varinjlim Y_n$.

Variants of (5) include: considering all diagrams of $Y_n$'s indexed by any directed poset, or by any filtered category. In the $\infty$-categorical literature, the term "compact" is used instead of "finitely-presentable".

If we require the transition maps between the $Y_n$'s to be injections, we get the notion of a

finitely-generated object.

A set is finite if and only it is quasicompact in the sense that the top element of its powerset lattice is finitely-presentable (equivalently, finitely-generated) as an element of that lattice.

The nlab says "compact" instead of "quasicompact" here, but I think in the original literature "quasicompact" is used -- if I recall correctly, the nlab has a general convention to never use the term "quasicompact", insisting instead that "compact" never implies "Hausdorff".

- A set $X$ is finite if and only if its coherent, i.e. it is "stably quasicompact" in the sense defined at the link.

(7) and (8) come from topos theory. There are corresponding notions in $\infty$-topos theory.

- A set is finite if and only if it lies in the closure of $\{ \{\emptyset\}\}$ under pushouts and intial objects.

Here are some other fun ones:

A set $X$ is finite if and only if all linear orders on $X$ are isomorphic.

A set $X$ is finite if and only if there exists a first-order theory $T$ such that $T$ has a unique model $M$ up to isomorphism, and moreover $M$ may be taken to have underlying set $X$.

countablyinfinite subsets" - trivially, a set is finite iff it has no infinite subsets. $\endgroup$ – Noah Schweber Jan 18 at 8:41isnothing more than an object of the category of sets; the axioms tell us something about what this sort of "set" is in just the same way that the axioms of ZFC tell us something about what its sort of "set" is. In Mike Shulman's terminology, there are "material set" theories like ZFC and "structural set" theories like ETCS. Both are adequate to support most of the ways that sets are used outside of pure set theory, so it's not unreasonable for mathematicians to differ on which sort of "set" is the default meaning for them. $\endgroup$ – Tim Campion♦ Feb 13 at 14:014more comments