There are two ways an offense can move the ball forward: By rushing or passing.
Rushing means that after the center snaps the ball to the quarterback, the QB will then turn and either hand or pitch the ball to another player—typically one who has lined up behind him. This player standing behind the quarterback, being handed the ball is a running back.
All of the offensive linemen will run forward, blocking for the running back who is free to run left, right, or up the middle of the field: wherever he thinks he can gain the most yardage. Remember, his goal is to try to score a touchdown.
Passing means that after the center snaps the ball to the quarterback, the QB will then throw the ball through the air to another player—often to a wide receiver who has lined up off to the side of the offensive line.
The QB is permitted to pass the ball to anyone except for the offensive linemen, so many teams will mix up who they throw the ball to. It could be the wide receivers (who do most of the pass catching), the tight ends, or even the running backs.