WebFinger is a protocol specified by the Internet Engineering Task Force IETF that allows for discovery of information about people and things identified by a URI. Information about a person might be discovered via an "acct:" URI, for example, which is a URI that looks like an email address.
WebFinger is specified as the discovery protocol for OpenID Connect, which is a protocol that allows one to more easily log in to various sites on the Internet.
The WebFinger protocol is used by the federated social networks GNU social, StatusNet and Diaspora to discover users on federated nodes and pods as well as the remoteStorage protocol.
As a historical note, the name "WebFinger" is derived from the old ARPANET Finger protocol, but is a very different protocol designed for HTTP.