The names of the halfback and fullback never made much sense to me but hopefully this can help:
-"Halfback" and "tailback" are synonyms. In fact, you'll never see "TB" on a roster. ("RB" or occasionally "HB" for halfback)
-A fullback may or may not be primarily a blocker. This is where that categorizational slang comes into effect. A power back, which can be either a RB/HB or a FB, has a role, and so does a blocking back (who will obviously be a FB).
A perfect example of the different roles of backs is the SB Tampa team of a few years ago. Pittman was the halfback, a speedy guy who could take it outside and catch out of the backfield. Alstott was the FB in the power back mould, a guy who was great at running up the gut and could lock for the passer. Cook, a blocking back in the truest sense, could lead block for either of the other backs or block for the passer.