Glad to be of assistance, brachiopod (still classifying fossils is Scotland?)
The Instrument Landing System (ILS) that electronically guides an appropriately equipped aircraft to a specific runway during inclement weather consists of several components, but to simplify the discussion, one needs to only consider the lateral (left and right) guidance called the localizer and the vertical guidance called the glide slope. The aircrat instruments presents this information in the form of a "cross hair" device (for the pedantic, yes I know the flight director is involved.)
In the Category III (actually there's IIIA, IIIb and IIIC as well as Cat I and Cat II), each with succedingly lower allowances for descent. The highest CAT I, has a Decision Height (DH = height above ground level at the runway threshold, usually) of 200 feet, down to essentially zero for CAT IIIC. Technically speaking, CAT IIIC has no DH since it's an autoland procedure, utiliizing an approved auto pilot. CAT IIIA allows descent to 100 feet at which time the pilot must have the runway environment in sight (let me tell you from experience, it happens really, really fast at about 130 knots airspeed or so). CAT III B allows DH of 50 feet...
If the runway environment isn't in sight a "missed approach" must be executed. It isn't uncommon for the main gear wheels to actually touch down during the missed approach procedure from CAT III of any class.
Most airports are not equipped for CAT III landings other than the major airports. Most are CAT I or at best CAT II...
These all are precision approaches but there are still non-precision approaches in use that use Minimum Descent Altitude (MDA) as the descent limits. These are usually around 500 feet or so and also require a minimum reported visibility, usually 1/2 mile... More thanyou ever wanted to know, I'm sure
Thanks for asking!