Only if a game supports it, which on PC, I can't think of any that would. I know some turn based games (the Civilization series, the Heroes of Mights and Magic series, etc.) support what's called a hotseat game. Since the game is turn based, the players can takes turn at the controls.
Most other computer games, such as shooters, simulations, etc. don't support this, because they're not programmed to play multiplayer that way. They're designed for each player to have a copy of the game and play over a LAN or the Internet.