Subgenres of comedy:
comedy of manners
comedy of character
romantic comedy
domestic comedy
situation comedy
satire, parody
farce
This list more or less moves from "high comedy" to "low comedy". High comedy tends to include much verbal wit, and, on the other end, most farce involves lots of physical comedy and is little dependent upon language. Most plays contain more than one style of comedy; however usually one dominates.

Aristotle on comedy: About "inferior" characters
presents us with ugly and ludicrous, not evil (morality is not the realm of comedy)
Aristotle briefly treats comedy in The Poetics, but it is often supposed that he wrote a second volume on comedy, which has been lost through the years.

Bonamy Dobree essay

"Critical Comedy"
moral defense of comedy
conservative reaction against change

"Free Comedy"
suspend laws of nature and society
morality irrelevant

"Great Comedy"
near to tragedy, but happy ending - absurd or cruel forces overthrown
serious themes - "disillusionment of mankind"

Why is great comedy less optimistic than tragedy? p. 455

Northrop Frye essay

Formulas ancient and yet still timely:
Move from society controlled by older, absurd or irrational order
To a society of youth, marriage, "right" order
End with celebration of new order - marriage, dance
Cognitio - comic anagnorisis or recognition
turns on mistaken identities, misinformation righted
Plotting: usually complex in comedy - absurd and comic in and of itself

Characters:
Blocking characters - parents or authorities
If the comedy focuses here, it's comedy of manners, character, satire, irony, or realism
Jonson describes these as "humors",
Bergson views repetition as funny
Ingenu characters - hero and heroine
Young, less developed and more of a blank slate, audience projects selves into them
If focus is here, it is a romantic comedy or domestic drama

Comedy incorporates as many as possible into new social order at the end
Essentially conservative
Audience must approve of the final, new society - joins in circle at the end

In comedy, "the social judgment against the absurd is closer to the comic norm than the moral judgment against the wicked." p. 465