étale morphism (submersion, immersion)
formally étale (formally smooth, formally unramified)
étale ∞-groupoid
in topology
local homeomorphism, étale space
étale groupoid
in differential geometry
local diffeomorphism, (submersion, immersion of smooth manifolds)
étale Lie groupoid
in algebraic geometry
étale morphism of schemes
étale site, étale (∞,1)-site
étale cohomology
There are several concepts of an immersion, depending on the category of spaces. The most basic ones are