Is it just another name for a running back or a wide receiver? It seems to me that it would be anyone who takes the pass from the QB and runs the ball which pretty much seems to be what the RB does. Is my understanding correct or is there more to it?
A pass rusher is on the defensive side of the ball (usually a defensive end but can be a linebacker in some defensive schemes) and his primary responsibility is to rush, or pursue across the line of scrimmage, the quarterback in attempt to disrupt the play.
Disrupting the play may be through sacking the quarterback, forcing the quarterback to hurry the throw, or forcing the quarterback to scramble. The less time the quarterback has to make a throw/play, the less likely it will be successful(1).
More information on the defensive rush can be found here.
A "rusher" is a running back from the offensive team who runs with, or "rushes" the ball, like you said.
But a "pass rusher" is someone from the defensive team who is "rushing" (chasing) the quarterback, who has elected to keep the ball for a (potential) pass.
This defender could be a lineman (end), linebacker, or even the "free safety" (from the backfield), who normally double covers the offensive team's best receiver, but was used instead in a "blitz."
You appear to have been confused by the similarity of the two terms.
A puss rusher is not someone who rushes with a pass but someone who rushes the passer or quarterback. NFL teams play two main styles of defense, named for their formations, the 4-3 and the 3-4. The first number refers to the number of defensive linemen and the second to the number of linebackers. Defensive linemen start each play crouched down in a modified sprinter's stance on the line of scrimmage. Linebackers start standing, usually right behind the defensive linemen. In the 4-3, the pass rushers are usually the outside defensive linemen or defensive ends. In the 3-4, the pass rushers are usually the outside linebackers.