higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A homomorphism of schemes $f \colon Y\to X$ is
finitely presented at $x\in X$ if there is an affine open neighborhood $U\ni x$ and an affine open set $V\subset Y$, $f(V)\subset U$ such that $\mathcal{O}_Y(V)$ is finitely presented as an $\mathcal{O}_X(U)$-algebra.
locally finitely presented if it is finitely presented at each $x\in X$.
finitely presented if it is locally finitely presented, quasicompact and quasiseparated.
essentially finitely presented if it is a localization of a finitely presented morphism.
A standard open $Spec(R[\{s\}^{-1}]) \longrightarrow Spec(R)$ (Zariski topology) is of finite presentation. More generally, an étale morphism of schemes is of finite presentation (though essentially by definition so).
The Stacks Project, section 28.22 Morphism of finite presentation
wikipedia: Finite, quasi-finite, finite type, and finite presentation morphisms
David Rydh, Why are unramified maps not required to be locally of finite presentation?, MO/206333/2503.