Abbey and Trappist
Trappist and Abbey beers aren't beer styles, per se: Trappist ales are brewed by monks or within the walls of a Trappist monastery and are certified by the International Trappist Association as having satisfied that and a few other strict requirements. There are a total of 178 Trappist monasteries worldwide (as of July 2015), but only 12 produce Trappist beer. Eleven of them are authorized to label their beers with the “Authentic Trappist Product” logo that shows they've complied with the strict rules of the International Trappist Association. Abbey beers, by comparison, are similar to Trappist ale flavour and style traditions with one big difference: any old John Smith could brew those beers and he doesn’t have to be a monk and give all his profits away to charity to do it.
Style-wise, Trappist and Abbey beers are generally described as Dubbels or Tripels, with some blondes or strong darks/Quadrupels thrown in for good measure. Of the 11 recognized Trappist breweries in the world that carry the ATP (Authentic Trappist Product) label, six are in Belgium (Westvleteren, Westmalle, Achel, Rochefort, Orval, and Chimay,) two in The Netherlands--one being Koningshoeven, one in Austria, one in Italy and one in The USA. On the other hand, it’s hard to quantify the number of abbey beer producers as, technically, any secular producer of the aforementioned styles are making abbey beers.