I can only speak for Canada, but assume it is the same worldwide.
The basic difference between the CAT's is simply the height to which you can go down to before executing a missed approach (if you dont have the runway visual). In most cases you can take the runway to mean the runway lights, lead-in lights, etc. If you see any of those, you can proceed below the Decision Height (DH)
To be designated as CAT III though, the equipment itself (the localizer and glideslope) must be certified for use beyond the threshold of the runway. This is what seperates a CAT I and II from III. In CAT I and II installations, the LOC and GS are not tested beyond and below the path past the threshold.
Now some figures.
A CAT I approach can be conducted down to 200' AGL (in most cases). Visibility must be RVR2600 (Runway Visual Range of 2600ft) or 1/2 sm.
CAT II approach is good from any height from 200' to not below 100', with a visibility from RVR 2600 down to RVR 1200.
CAT III is dependent upon the installation, but can have DH's below 100'. There are also 3 different types of CAT III installations, A, B and C of which your normal instrument rating only covers A. B has more stringent heights and visibilities and CATIIIC is strictly Autoland only (pretty much 0 Visibility, 0 Cloud height)
I hope this helps!