Fullback is usually a lead blocker for the halfback or tailback. Halfback and tailback are the same thing. Fullback is usually only a ball carrier on short yardage or in the option. In the option offense you will see 3 backs in the back field one is the fullback, second is the upback, then the half or tailback.
The position names can indeed be confusing. As a general rule, the name relates to the relative distance from the line of scrimmage to the rear of the backfield formation the player sets up, with the QB being 1/4 of the distance, the HB being 1/2 of the distance and the FB setting up at the full distance. A TB (tailback) also brings up the rear of the formation set. A WB (wingback) sets up to the side of the formation.
This isn't always true, as modern formations have moved the positions around at times. As an example, the spread option (which we run), uses the TE (someone who usually sets up as the end man on the line of scrimmage) halfway back (but set to the side) in the backfield as a lead blocker and calls the position either TE, 3-Back or H-Back. This position could also be called a slotback, but slotbacks are usually the size and speed of HBs/TBs.
The HB position was prevalent in the days of T-formation football, where one was used on each side of the formation. Their job was primarily to take a hand-off from the QB and either carry the ball or be a decoy and/or block for the ball carrier. The FB's job was usually as lead blocker, but was sometimes a ball carrier.
The FB was also used in I-formation, with a TB set up behind him, which goes against the distance terminology.
The RB (running back) term is a catch all descriptor that would include HB, TB and FB.
It can definitely be confusing, but it gets worse when you consider all of the different offenses. Today's GaTech game will feature an A-back/slotback and a B-back. Both are RBs, but you could say one is a TB or HB/slotback and the other is a FB.
Clemson's backfield usually consists of a QB, a TB/RB and an H-back/3-back/TE/slotback.