The differences between running backs is based on their:
A half back is a running back that is positioned close to the quarterback on running plays. They take the hand off on running plays and are faster and smaller than the fullback. He is the ball carrier for most of the plays. Many people would argue that the half back is one of the most important roles for success in a game.
A tailback is a running back similar to a half back. Tailbacks are positioned behind the quarterback forming a T shape on the field. They are responsible for catching the ball and running the ball. This player must be a very versatile player and willing to fill in where needed for the given play. He will often catch the ball when he's in the backfield.
A fullback is a running back that is positioned close to the half back or tailback at the start of the down. Fullbacks are bigger than normal running backs. The fullback defends the ball carrier from being tackled. He is one of two running backs. His main role is to protect the halfback. This player must be quick on his feet and very strong to fend off opponents. He should clear a path for the halfback to get the ball through and down the field.
On a passing play the running back will get open on the field for pass thrown by the quarterback. The running back may be responsible for blocking on passing plays as well, making sure the quarterback has extra time to throw to one of his receivers. Running backs are very multifaceted players, who need to be resourceful in different ways depending on the play.
On a running play the running back (half back or tailback) will get the ball from a hand off from the quarterback. The fullback will protect the ball carrier (half back or tailback) from being tackled by defensive players. Because of this, the fullbacks should be very strong and aggressive, as they are literally moving multiple defenders out of the way to create space for their teammates.