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Immediately behind the quarterback is the fullback. He has two jobs: if it's a running play, the full back will hit the hole at full speed immediately ahead of the running back and will push any remaining defender out of the hole, making certain it's clear for the running back. If it's a pass play, the fullback's job is to stay near the quarterback and block any defenders that happen to make it past the offensive line. The fullback is a fairly quick and fairly big guy, typically around 6' to 6'2 and weighing around 250 pounds.
Sometimes if a team has a very athletic fullback type, they will put him in motion. This means as the quarterback is calling out signals, before the ball is snapped, the fullback will suddenly stand up and start running. When a player goes in motion he may run to the left or the right, but he may not run towards the defense until the ball is snapped. By putting the fullback in motion, you challenge the defense to decide who’s going to guard him. At the snap of the ball, the fullback will turn towards the defense and either hit a key defender or run a bit to catch a quick pass. If the fullback is put in motion frequently, then he's sometimes called an H-Back. When he's used as an H-Back, the fullback may start out lined up just outside the left tackle as if he were an extra tight end, but then go into motion and play from behind the linemen as if he were a fullback.
This is one of the two most important people on the offense. The running back has several important jobs to do. If the quarterback calls a running play, he will hand the ball to the running back. The running back then has the job of finding a hole in the defense and trying to get as far as he can. If the quarterback calls a passing play, the running back will often have the job of staying near his quarterback for a second or two and blocking any defensive players who make it past the offensive line. Finally, the running back will often be sent out as an additional receiver.
There are several types of running back in the NFL. Some of these guys are somewhat larger and stronger and are called "straight-ahead" or "downhill" backs. Jerome Bettis of the Steelers made much of his yardage running straight ahead, right at the center of the line. It was the defense's job to get enough large guys in his way to stop him. When a defender did get in Bettis' way, Bettis would deliver a crushing hit on the defender, making the defender think a bit before getting in Bettis' way on the next play.
Some of these guys are very quick at changing direction and exceedingly fast in a straight line. Terrell Davis of the Broncos made his living as a cutback runner. He would run one direction for a second or two, then when the entire defense was moving that direction to try to keep up with him he would suddenly change direction, perhaps even back the way he can come from. Few defenders had the speed to keep up with him.
Some of these guys have some miraculous ability to twist and turn a bit as they run, eluding defenders one after another seemingly by a couple inches each. Barry Sanders of the Lions made his living doing this. He would have 50 yard touchdown runs where it seemed every single guy on the defense had his hands on him at one time or another, but none of them could bring him down. Barry used to say in interviews, "I don't even know where I'm going next, how could the defense possibly know?"
There are also all-around backs. Marshall Faulk of the Rams would make as many yards each year as a receiver as a runner. He was an outstanding player all over the field - all you had to do was find a way to get the ball into his hands, and his speed and competitive spirit would take over. Where ever he lined up on the field, as a running back, in motion before the play started, or on the line as a receiver, you had to account for him with your defense as if he was the star of the coming play. If you didn't, the Rams would find a way to get him the ball and he would make a big play. If you did, often the result would be a hole in your defense somewhere else that the Rams would find a way to exploit.
Running backs come in all shapes and sizes, from the larger backs like Jerome Bettis at well over 6' tall and about 250 pounds, down to as small as 5'6" and 160 pounds. A good running back will touch the ball 20 to 30 times each game, getting hit solidly by the defense on most of those plays. So these guys tend to have relatively short NFL careers. It is unusual to find a running back who is still playing at a high level after he is 30 years old. The good news is they get paid very well, as much as $8M to $10M per year for the best of them.