The person who was famous for his comedies was Aristophanes
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Aeschylus.
Aeschylus was a Greek tragedian.
The only ancient Greek comedian whose works survive is Aristophanes; eleven of his 40 or so plays remain. He wrote plays poking fun at contemporary life and famous figures at the time. His play "The Clouds" made fun of Socrates, and the satire was so vicious that Plato claimed the play caused Socrates to be executed. Much of Aristophanes' humor is crude, even disgusting, but much of it is still funny today; for example, in "The Birds", he lampoons lawyers and the tendency of ancient Athenians to sue over even the smallest insult.
Why ancient Greek comedians of course! buh-dump psh! Actually it is that simple, unless you want names. If so, Aristophanes and Menander are pretty good. However, this is largely pretty mature stuff. Aristophanes wrote a play telling how the Peloponnesian War between Athens and Sparta ended, the women of Greece collectively withheld sex from their husbands (even the prostitutes stopped, and that was a pretty heavy business since most men didn't marry until their late 20s, thirties). Some of Aristophanes' stuff was so raunchy that it wasn't properly translated for a while. Naturally, I highly recommend Aristophanes.
In his works, Aristophanes considered Socrates and Euripides to be his soulmates.