halfback - the halfbacks may function as running backs, blocking backs or short-yardage receivers. In some formations, running back positions may have specialized names. In a traditional formation, there are two halfbacks. Modern formations typically have only one, and some formations have none. However, the term has all but disappeared from the modern American football vocabulary, replaced by "running back". fullback - positioned behind the middle of the line, a fullback may do some running, some blocking, and some short receiving. A classic fullback is more of a straight-ahead, "four yards and a cloud of dust" power runner than a halfback. Many modern formations do not use a fullback. Most plays utilizing the fullback call for him to block, generally by running up the middle of the line, clearing a path for a running back to use. tailback - a player positioned behind the middle of the line and deepest of all backs. source: http://www.explore-football.com/football/O/Offensive_team.html

The terminology for the offensive backfield goes back to the origins of football, and describe where the players originally stood in the formation. The "full" back was fully back, or the furthest back in the offensive backfield. The "half" back was halfway between the fullback and the offensive line. The "quarter" back was a quarter of the way back. Depending on the offensive formation, especially in modern football, the fullback may be in front of the halfback, despite the name. Other names and positions, which vary from play to play, include: Tailback, T-Back, Fullback - the "tail," or deepest back, commonly directly in line and behind the Center and Quarterback. Running Back, H-Back, Halfback - technically describes any back in the backfield, but commonly refers to the remaining two offensive back positions (after QB and FB.) Sweep Back, S-Back - describes a player, usually lined up wider to the left or right, who will often run a sweep with the ball, or execute a fake sweep to distract the defense. Wing Back - A wing lines up just outside, and a step back, from a Tight End. Also called a Flanker. Slot Back or Slot Receiver - When the End is Split from the Tackle, instead of playing Tight, there is a slot between the end and the tackle. The Slot Back lines up in the slot, and a step back. Depending on the play, he might be a running back, or a receiver. There are multiple terms at play. Some describe the position, and others describe the responsibility. For example, if the Wingback or Slot back (describing position) takes a sweep path, he is a Sweep Back (responsibility.)

Also note that many teams invent their own names for the running back position. At the University of Florida the running back is often referred to as the gatorback, scatback, or scrapback.