What are the differences between each ILS category?
The category specs are easily found on Wikipedia. Here is a concise view of the ILS approach minimums, as far as the crew is concerned.


  Cat I
  
  
  DH: 200 ft (61 m) or more
  RVR: 1,800 ft (550 m); at some airports 1,210 ft (370 m) is approved. For single crew operations, increased to 2,600 ft (790 m)
  orVISIBILITY 800 m (2,600 ft)
  
  
  Cat II
  
  
  DH: less than 200 ft and more than 100 ft (30 m)
  RVR: 1,000 feet (300 m)
  
  
  Cat IIIa
  
  
  DH: less than 100 ft and more than 50 ft (15 m)
  RVR: 600 feet (180 m)
  
  
  Cat IIIb
  
  
  DH: less than 50 ft (15 m) or none
  RVR: 150 feet (46 m)
  
  
  Cat IIIc
  
  
  DH: No limitations
  RVR: None
  


They are based on decision height and RVR (or visibility for cat I).
Note that there are no public transport aircraft certified for Cat IIIc as there is no use for landing without visibility and then staying on the runway because you cannot visually taxi to the gate.

The requirement for equipment certification accuracy itself can be found here.