There are 2 fully functioning independent autopilots on the A320. If the captain is flying, he engages AP1 when he wants the aircraft to fly itself; the first officer would engage AP2 when he is flying. For most phases of flight, both cannot be engaged at the same time - pressing the other button would just switch systems. When flying the approach phase it is possible to engage both autopilots and then the aircraft is capable of an autoland.
The category of ILS (Instrument Landing System) approach - CAT I, II or III - is dependent on visibilty and cloud base and have different heights at which you need to see the runway before continuing to land. A high percentage of approaches are flown down to CATI minima before deciding whether to continue and land manually or "go-around".
If the weather drops below a certain value CAT II and CAT III approaches will be flown. Some airports and some aircraft are only capable of CATII approaches; the airbus can fly a CATIII approach. Basically this means the aircraft can get closer to the runway before you need to see it or the lights to continue and land.
In our company, if we are using the autopilot, we do engage both of them even for a CATI approach, even though only one is needed. This is to do with redundancy - if one AP suddenly fails the other one is there to take over instantaneously. When the flying pilot is happy and ready he will disengage them both and continue flying manually.
We can also do practice autolands - to keep ourselves current for when we need to do one for real. In this case, even on a bright blue day, both autopilots will be engaged on the approach and, even though it is a CATI approach, the aircraft will fly and land itself.
It is a bit confusing but basically CAT I means one autopilot and a manual landing, CAT II or III means two autopilots and an autoland.