Trappist and abbey beers often sit side by side on beer menus. They look alike, but there's a clear difference.
Trappist
A Trappist beer is brewed in or near an abbey of the Trappists, a monastic order, under the supervision of the monks. The proceeds go to the abbey and to charity. Only beers that meet these rules may carry the 'Authentic Trappist Product' logo of the International Trappist Association.
Well-known Trappists include Westmalle, Orval, Rochefort, Chimay and, from the Netherlands, La Trappe.
Abbey beer
An abbey beer refers to an abbey, but is brewed by a regular brewery, often under licence. Sometimes the abbey still exists and receives money for it; sometimes it's mainly a name with history. It says nothing about quality: there are very good abbey beers.
Examples are Leffe, Affligem and Grimbergen.
And on the menu?
Trappists are often a bit more expensive than abbey beers. In BierChecker you can see per beer where it's cheapest, so you can choose for yourself.