In a comedy, especially a slapstick comedy, a "gag character" is a character with one exaggerated personality characteristic, used for comedic effect. Often this gag characteristic is the basis for some rather broad comedy, especially if it is based around something like being particularly hungry, horny or greedy.
Gag characteristics can vary from the simple (Wimpy from Popeye likes eating hamburgers!) to the high concept (David Chapelle's Clayton Bigsby is a blind black man who is racist against black people). William Shakespeare created a gag character, Sir John Falstaff who was so popular that he later got a play centered around him.
Often, in situation comedies, characters would go from being gags to being serious, depending on plot focus. In classic 80s comedy Night Court, the characters would go from one note personalities: "The Wacky One", "The Horny One", "The Weird One" and "The Innocent One", during background moments, to being more fully developed characters when the story was about them.