A finite integral domain is a field.

Proof:
Let $R$ be a finite integral domain. Let $a$ be nonzero element of $R$

Define a function $\varphi \colon R \rightarrow R$ by $\varphi(r)=ar$

Suppose $\varphi(r)=\varphi(s)$ for some $r,s \in R$ Then $ar=as$ which implies $a(r-s)=0$ Since $a \neq 0$ and $R$ is a cancellation ring, we have $r-s=0$ So $r=s$ and hence $\varphi$ is injective.

Since $R$ is finite and $\varphi$ is injective, by the pigeonhole principle we see that $\varphi$ is also surjective. Thus there exists some $b \in R$ such that $\varphi(b)= ab = 1_R$ and thus $a$ is a unit.

Thus $R$ is a finite division ring. Since it is commutative, it is also a field.

Note:
A more general result is that an Artinian integral domain is a field.