I will assume that you mean the dramatic genre tragedy when you write drama.

Both the tragedy and the comedy were born around 500 B.C in Athens and the two genres played different roles in the greek society.

The tragedy grew out of the cult for Dionysos, the deity for ecstacy, wine and fertility: men would honor the god dressed as goats (associated to Dionysos' Tragos) accompanied by song. From this grew the antique tragedy, and so it always had a more solemn tone to it, as it was a part of worship.

The purpose of the antique greek tragedy was to "cleanse" the viewers through catharsis. The tragedies always ended with the main characters' deaths as punishment for some crime against the gods, most notably hubris. It was thought that when the main character realized his wrongdoings and got his punishment, the audience would experience a psychological cleansing effect (catharsis).

The comedy was a more "common" form of art, but it had its place right next to the tragedy. During great festivities lasting up to three days, five plays would be performed a day, of which at least one was a comedy. The catharic element that was missing from the comedy as well as the more profane nature of it made it a bit less popular than the tragedy in ancient Greece. Nevertheless, great comedic writers have stood the test of time, such as Aristofanes, who rightly can be called the master of the greek comedy.

My sources are two swedish textbooks on the history of literature mainly in Europe: Språkets värld (Abrahamsson, Mentzer-Ekholm, Schybergson, Tuominen) and Bokskogen (Mentzer-Ekholm, Rintamäki-Johansson, Schybergson, Söderling).