Finite morphism

A morphism $f:Y\to X$ of schemes is finite if every affine open $U=\operatorname{Spec}A\subseteq X$ has affine inverse image

and $B$ is a finitely generated $A$-module through the induced ring homomorphism $A\to B$.

It is enough to verify this condition on an affine open cover of $X$; finiteness is affine-local on the target.

For a field extension $K/F$, the morphism $\operatorname{Spec}K\to\operatorname{Spec}F$ is finite exactly when $K$ is finite-dimensional over $F$, that is, when $K/F$ is a finite field extension.