Link: http://yarchive.net/env/jet_fuel.html

There are really only two types of Jet fuel used worldwide, "narrow cut " Jet-A1 which is used for almost all civilian aircraft, and Jet-B, the "wide-cut" (includes some of the gasoline fraction as well as the kerosine fraction) fuel used by the military.
Most engines are designed to handle both, and the varying specification names are mainly historical, however ( AFAIK ), all civilain passenger aircraft use Jet-A1 worldwide. During the last two decades the only divergence has been in the changes of freezing point, as the original -50C maximum specification was really only required for a few high altitude long-distance routes ( eg over Himalayas ), and that a relaxation to -47C would be acceptable, with the ocassional need to modify flight plans to avoid cold air. That reduction was
estimated to increase the global availablity of Jet-A1 by almost 10%, thus keeping the prices down. There is now also provision in the Jet-A1 specification for carriers and suppliers to negotiate the freezing point.
Jet A is similar to Jet A-1 except that the maximum freezing point is -40C, and so it is not commonly used internationally.
There are a large number of specification that are variants of Jet-A1, with anti-icing additives, static dissipators, and ( more recently ) differing freezing points for some regional products, but they are all essentially Jet-A1 and are compatible with each other.
The military also have special versions of kerosene. The widecut specification was originally designed to maximise yield of jet from crude oil ( up to 40% ) which could be in short supply during wars, and also to have excellent low temperature properties ( -60C maximum freezing point ) for global operational use. There are other military kerosenes, eg for safety reasons, aircraft carriers use a high flash point fuel, minimum FP=60C, versus 38C for Jet-A1. There is no flash point limit for Jet-B, but it will be much lower.

Essentially today there are only two common types, Jet-A1 and Jet-B, each of which is covered by multiple specifications.

Jet-A1 = AVTUR = NATO F-35, and ( with slightly differing static dissipating additive related specifications ), = AVTUR/FSII, Nato F-34, US JP-8. Jet A is identical to Jet A1, except for freezing point specification.

Jet-B = AVTAG/FSII = Nato F-40 = US JP-4

The less common high flash aircraft carrier fuel is AVCAT/FSII=Nato F-44=US JP-5.