To add to what's already posted: CAT III is sub-divided into IIIa, IIIb, and IIIc. These refer to the decision height / visibility at the effected airport, as well as the aircraft. For example: Delta's MD11s are CATIIIc, because they have enough redundant systems and power sources to perform an auto-land in Zero / Zero weather (0 visibility at 0 altitude). If, say, one of the three autopilots were to fail, the aircraft would be restricted to CATIIIb landings untill the problem was repaired and the aircraft sucessfully completed a full auto-land (in good weather). Any modern autopilot is capable of performing at least CATIIIa, but the aircraft may be a limiting factor. When I first hired on at Delta, we had two 737-200s leased from Guiness Pete (EI-BON and EI-BOM, if anyone cares). They were limited to CAT III a due to a lack of autopilot feed back input instrumentation, among other things.
CAT IIIc also refers to airport equipped with runway, taxiway, and gate 'guidance systems'. So far as I know, only Heathrow is CATIIIc active. This system consists of a wire buried in the runways, taxiways, and ramps, which act as an antenna for the signal telling the aircraft where on the airport it is. Technically, the aircraft rolls out, turns off, and taxis right to the gate in 0/0 conditions under autopilot commands. All the pilot has to do is set the brakes and shut down the engines!