Class A
Trips between 4 and 6 mA. The people-protection device: every receptacle, breaker, plug head and in-line module for 120 V and 240 V branch circuits. When the Code says "GFCI protection for personnel" it means Class A.
Class B
Trips above 6 mA and up to 20 mA. Written into UL 943 for one purpose — underwater swimming-pool lighting installed before the 1965 edition of the Code, whose fixtures leak too much for a Class A device to hold. You will not meet a Class B device outside that application, and you should not install one anywhere else.
Special-purpose GFCIs
UL 943C covers ground-fault protection for equipment on circuits Class A cannot serve: 480 V and 600 V three-phase, welders, large chargers, mining and industrial gear. Classes C, D and E trip at 20 mA and are paired with a monitored grounding conductor, because at those voltages the shock hazard is managed by keeping the frame grounded rather than by tripping at 5 mA. NEC 210.8 does not call for them in dwellings; they appear in industrial specifications and in Article 590 temporary power at higher voltages.
Not a GFCI class: GFPE
Ground-fault protection of equipment (NEC 215.10, 230.95) trips at hundreds of amps to save a service or feeder from burning down; it protects copper, not people, and is often confused with the small devices on this site.
The portable ones we build
Our plug-head and in-line GFCIs are Class A, UL 943 listed, 120 V and 240 V, in manual- and automatic-reset versions (which reset). Our export cords carry 30 mA and 10 mA RCDs to the IEC standards instead (RCD cords).