It is a circular metal (usually cast iron) plate which fixes to the end of the lathe spindle.
The faceplate may be attached to the lathe in several ways: the two most common are a thread and a precision cone arrangement, and threaded studs and a circular recess fitting a flange on the end of the spindle.
Increasingly common is the camlock arrangement, in which shaped studs and cams replace threaded studs for rapid exchanging of the faceplate with other accessories, such as three or four jaw chucks.
The smaller plate in the upper photo is specifically a 'dog face' with slots intended to hold a bent tail dog while the work itself was supported on centers.
It may seem that a faceplate is a primitive accessory superseded by precision chucks, but its inherent flexibility (almost any shape can be attached to a faceplate with care and the right fixings) and the possibility of achieving great accuracy by careful setup make it essential for a well equipped lathe.