Any rushing player is called a rusher.
Rushing, on offense, is running with the ball when starting from behind the line of scrimmage with an intent of gaining yardage. While this usually means a running play, any offensive play that does not involve a forward pass is a rush - also called a run. It is usually done by the running back after a handoff from the quarterback, although quarterbacks and wide receivers can also rush. The quarterback will usually run when a passing play has broken down - such as when there is no receiver open to catch the ball - and there is room to run down the field. A team with a quarterback who is fast and skilled at running may regularly call intentional running plays for that quarterback, but this is rare due to the increased risk of injury. A wide receiver can act as a rusher on several kinds of plays, such as on a reverse, on an end-around, or on a lateral pass behind the line of scrimmage, which is a type of screen pass. However, a wide receiver screen play is usually intended to be a forward pass so that if the receiver drops the ball it is an incomplete pass instead of a fumble.
A rushing attempt may also be referred to as a carry, with any yards gained referred to as rushing yards, as in "the running back had 20 carries for 100 rushing yards."