Let \(\pi \colon X \to Y\) be a morphism of schemes.
\(\pi\) is finite if and only if it is integral and locally of finite type.
If \(\pi\) is finite, then it is quasi-finite: every fibre is a finite discrete set [@Har10a, Exercise II.3.5].
If \(\pi\) is finite, then \(X \cong \operatorname{\mathbf{Proj}}_Y \mathcal{S}\) over \(Y\) for a quasicoherent graded \(\OO_Y\)-algebra \(\mathcal{S}\) generated in degree \(1\) by the finite type module \(\mathcal{S}_1\); in particular \(\pi\) is projective in the sense of D-SCHRELSPECPROJ, and proper.
A morphism of locally Noetherian schemes is finite exactly when it is proper and quasi-finite, equivalently proper with finite fibres [@Har10a, Exercise III.11.2]. Without Noetherian hypotheses: finite exactly when proper, locally of finite presentation, and with finite fibres.
This characterization converts the module-theoretic definition of finiteness into geometric conditions. The implications are \[ \text{finite} \implies \text{projective} \implies \text{proper} \implies \text{universally closed}, \] with none of them reversible: \(\PP^1_k \to \Spec k\) is projective and not finite, and complete non-projective threefolds give proper morphisms to a point that are not projective.
The converse direction, proper plus finite fibres giving finite, recognizes normalization maps and proper zero-dimensional-fibre morphisms as finite.