Before the IBC, we had the UBC on the west coast. IBC's B's can be considered to be similar to UBC's N's (which meant Not rated), A's generally represent fire rated construction (i.e. 1-hour).
Careful Wireman...it is possible to utilize sprinkler protection in lieu of the fire-resistance rating of "A" vs. "B", but Exception 2 of 2003 IBC Section 506.3 does not allow an increase in area on top of sprinkler substitution for fire protection.