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