These are quoted from Federal Aviation Regulations section 1.1:
Category: (1) As used with respect to the certification, ratings, privileges, and limitations of airmen, means a broad classification of aircraft. Examples include: airplane; rotorcraft; glider; and lighter-than-air; and (2) As used with respect to the certification of aircraft, means a grouping of aircraft based upon intended use or operating limitations. Examples include: transport, normal, utility, acrobatic, limited, restricted, and provisional.
Class: (1) As used with respect to the certification, ratings, privileges, and limitations of airmen, means a classification of aircraft within a category having similar operating characteristics. Examples include: single engine; multiengine; land; water; gyroplane; helicopter; airship; and free balloon; and (2) As used with respect to the certification of aircraft, means a broad grouping of aircraft having similar characteristics of propulsion, flight, or landing. Examples include: airplane; rotorcraft; glider; balloon; landplane; and seaplane.
Regulations, by their nature, are over-precise and overexplanatory. I'm bringing up these two because they are at the core of a lot of FAA regulations, and because there's an inherent contradiction in them. "Airplane" is a category with respect with pilot certification, but a "class" with respect to aircraft. This just one of those things you have to learn to be a pilot.
Starting to study for the FAA written, can you tell? :-)