Like somebody before correctly pointed, you have to segregate CAT 1, 2, 3 from the autoland capability an aircraft can have.
Cat I can be flown manually or autopilot coupled down to minimums. The landing can be manual (most cases) or automatic (if the aircraft is able), but some restrictions apply because the CAT I ILS signal accuracy is not guaranteed below CAT I minima, therefore closer attention has to be paid by the crew in case they elect to do an autoland.
Cat II has to have an autopilot functional in order to start flying the approach. If it subsequently fails , you can still continue, it just depends where it fails on you, but this is another discussion.
CAT II can be landed manually or autoland. This is legally speaking. Some companies elect to prohibit manual landings from a CAT II approach, others simply cannot do that because their aircraft do not have autoland capability. (Ex: most regional jets, ATR's, SAAB's, EMB).
CAT IIIA- You have to have 2 (or more) autopilots functional AND autoland capability. For some aircraft (and I can offer you the example of the SAAB 2000) you can have a Head Up Display instead of the autoland capability, this is the only case when you can manually land the airplane following a CATIIIA. Otherwise, autoland.
CAT IIIB - You have to have autopilots and autoland capability.
The discussion can go further into if the autopilots are fail passive or fail active, different company's procedures and so on, but it would be too long.
Ref:JAR TGL 23, jar awo, etc