An engine is any machine or computer program that plays chess.
Chess computers are frequently broken down into two pieces - the user interface and the actual chess engine.
With respect to this web site, the user interface isn't so important. It can be a chess board that detects where the pieces are, a web page that allows pieces to be moved, a command line that allows moves to be typed in, or any number of other possibilities.
The engine is what selects the moves on behalf of the computer, and is thus where matters of chess-specific interest lie. Some engines commonly employed on personal computers include Stockfish, Houdini, Fritz, Crafty, Rybka, and Fruit. In 1997, IBM's Deep Blue became the first engine to defeat a sitting world chess champion, Garry Kasparov, under classical match conditions.