(Finite type)
\(f : X \to Y\) is locally of finite type if \(Y\) has an affine cover by \(\Spec B_i\) such that \(f^{-1}(\Spec B_i)\) has an affine cover by \(\Spec A_{ij}\) with each \(A_{ij}\) a finitely generated \(B_i\)-algebra. It is of finite type if in addition each \(f^{-1}(\Spec B_i)\) is covered by finitely many of the \(\Spec A_{ij}\), that is, locally of finite type and quasicompact.
(Finite presentation)
\(f\) is locally of finite presentation if each \(B_i \to A_{ij}\) is of finite presentation: \(B_i[x_1, \dots, x_n] \surjects A_{ij}\) with finitely generated kernel. It is of finite presentation if it is also quasicompact and quasi-separated.
Finite type is a hypothesis of the valuative criterion of properness and part of the definition of a proper morphism. Over a locally Noetherian base \(Y\), locally of finite type and locally of finite presentation coincide, because every ideal of \(B[x_1, \dots, x_n]\) is finitely generated for \(B\) Noetherian; for \(Y\) Noetherian, finite type and finite presentation coincide. Over the non-Noetherian ring \(B=k[x_1,x_2,\ldots]\), the quotient \(B\to B/(x_1,x_2,\ldots)\cong k\) is of finite type and not of finite presentation. Finite presentation is the finiteness condition in the definition of smooth and étale morphisms and in limit arguments.