higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
An Artin stack is a geometric stack over the étale site: the stack quotient of a scheme whose automorphism groups are algebraic groups.
This is the general notion of algebraic stack. But often the latter term is used in a more restrictive sense to mean Deligne-Mumford stack.
Artin stacks are named after Michael Artin.
See for instance
Gérard Laumon, L. Moret-Bailly, Champs algébriques , A series of Modern Surveys in Mathematics vol. 39, Springer-Verlag 2000.
Dennis Gaitsgory, Nick Rozenblyum, A study in derived algebraic geometry, Section I.2.4.