The term "pinch hitter" comes from baseball as the batter is expected to score some quick runs and relieve the pressure without having much concern of losing the wicket.
Role
The term is used to denote a batter who is promoted up the batting order. It is often done in limited overs cricket to increase the rate of scoring and relieve the pressure while chasing a big target. Often pinch hitters aren't expected to stay for too long and hence, are usually lower orderbatters or all rounders whom the batting side can afford to lose in the effort to score quick runs. They might come in instead of the regular designated batter typically at a critical points in a match.