Anime whose central struggle causes hilarious results. These stories are built upon funny characters, situations and events. A comedy anime is laced with humour and sets out to provoke laughter from the audience.
Japanese humour can be a bit strange to westerners, so if you're new to this type of humour, just bear with it; it'll most likely grow on you if you're a fan of other kinds of comedy.
Comedy — Subcategories
Anime that imitate other stories (can be from TV, film, books, historical events, ...) for comic effect by exaggerating the style and changing the content of the original. Also known as spoofs. It can also refer to when whole elements of one work are lifted out of their context and reused.
Slapstick humour is generally recognized by lots of physical abuse, horseplay and practical jokes. Also often seen are impossible situations, using sounds to illustrate impossible stunts that aren't shown on-screen and to create tension for audience anticipation. (Source: Wikipedia:Slapstick)
Violent Retribution for Accidental Infringement
A comedy / slapstick element often found in Ecchi anime: The hilarity derives from spontaneous embarrassment, overreaction of the "victim" (girl), immediate retribution for a seemingly inappropriate act, and a sense of pity with the supposed "aggressor" (boy).
Typically a boy will - through no fault of his own - stumble into a girl, thus invading her personal space. In the process the boy may e.g. pull down her skirt, grab the girl's breasts, or accidentally kiss her. Any level-headed girl would see the accident just for what it is. Not so the girl with an overly developed sense of modesty, and a thing for violence. She - often a tsundere - will clobber the boy with excessive brutality.
A caricature style often seen in anime when the character shows extreme emotions, like fear, anger, happiness, surprise, or shame. Seen as cute, as part of humorous comedy. The character's face suddenly turns into a simplified version, the body will usually also deform into that of a small child (oversized head, short limbs, extra-large tears). (Source: Wikipedia:Super deformed)