What is a REST API?
An API that follows REST conventions, using HTTP methods on resource URLs to model create/read/update/delete operations.
REST (Representational State Transfer) is an architectural style described by Roy Fielding in 2000. The relevant principles for everyday work: resources have URLs (/users/42), HTTP methods describe the operation (GET to read, POST to create, PUT/PATCH to update, DELETE to remove), and the server returns a representation of the resource, usually JSON.
In practice, "REST API" today means "JSON over HTTP that uses verbs sensibly." Strict Fielding-style REST with HATEOAS hyperlinks is rare; most real APIs are JSON-RPC dressed up in REST clothes, and that's fine. The conventions are valuable; the dogma is not.
REST won the API war in the 2010s by being radically simpler than SOAP. GraphQL and gRPC have eaten chunks of its territory since, but for public APIs that need to be approachable from any language, REST is still the safe default.
In the wild
GET /users, list usersPOST /userswith JSON body, create a userDELETE /users/42, remove user 42