There some very famous plays and playwrights of ancient Greece. Sophocles is the author of probably the most famous tragedy of all time: Oedipus Rex. This play is part of The Oedipus Trilogy, which also includes Antigone and Oedipus at Colonus. The entire cycle dramatizes the story of Oedipus, how he fell prey to the oracle that predicted that he would kill his father and marry his mother, and what happened to him and his family after the moment that he realized that this prophecy had been fulfilled.
Aeschylus is another famous author of tragedies. His Oresteia is a collection of plays about King Agamemnon and his family. The specific play titles are:
Agamemnon, Choephoroi (The Libation Bearers), and Eumenides (which refers to the “kindly ones,” the avenging furies who seek vengeance on Orestes). [from Enotes Greek Drama Study Guide]
Euripides, also an author of tragedies, wrote the famous play Medea. This work examines what happens to Medea when Jason (the Jason of myth who is associated with the Golden Fleece) rejects her for a new wife.
Comic plays that are famous are Lysistrata, Frogs, and Birds, all by the most famous playwright of Greek comedies, Aristophanes.
For more on Greek Drama please follow the links below and consult the Greek Drama Study Guide.