Trappist & Abbey Ales
"Trappist" is an appellation (i.e. a legal description with the enforceability of a trademark) meaning that only brewers who meet certain requirements are allowed to use the Trappist name.
For a beer to have the Trappist designation, it must be brewed under the supervision of monks at a brewery on the monastic property.
Abbey ales are similar styles to Trappist ales but are brewed by secular commercial breweries that may be under license to an active abbey, named for a defunct abbey, or without any monastic name connection at all.