In Ancient Greece, people wrote for public performances. An example of this is theater. There were three main types of theater productions in Ancient Athens: satyr plays and tragedy and comedy. We aren’t exactly sure how these genres of drama developed or come to be important to the Ancient Greeks, but Aristotle and a number of other writers wrote about and proposed theories of the origins of theater. What we do know is that, Greek plays were produced in an outdoor theater with three actors and a chorus. Masks were used (instead of elaborate costumes). And, at least in the beginning, the plays were performed as part of a festival to honor the god Dionysus. They were usually only performed once, and that was in a contest during this festival.
Originally, the Cult of Dionysus would honor him with satyr plays, a form of tragicomedy similar to what we could call burlesque. The chorus would be mythological satyrs and the play would focus on a mythological theme. The plays, though, were really about crazy drunkenness and sexual innuendos since that is what Dionysus was known for. Tragedy was the genre of play that dealt with the mythological past and the tragedies from those stories or origin. Comedy was almost always about contemporary public figures and issues (a kind of forum for making fun of public figures and public concerns; think Stephen Colbert).
The major components of the theater are the orchestra, theatron, skene, and parados. The orchestra was a circular dancing space in which the chorus would interact with the actors. The audience sat in the theatron, probably with homemade seats (pillows, cushions, etc.) for several centuries. The skene was directly behind the stage and was a tent decorated as a palace, temple, or other building for the appropriate backdrop of the play. The parados were the passageways or paths for entrances and exits.
Important names to know in Ancient Greek Theater are Aeschylus, Euripides, Sophocles, and Aristophanes. The first three are the writers of the only surviving tragedies from that time period. According to Aristotle, Aeschylus (about 524 BCE), transformed Greek theater by expanding the number of actors required. He was most known for Seven Against Thebes. Only seven of his estimated seventy to ninety plays have survived. Euripides (about 480 BCE) wrote Medea. Ancient scholars attributed 95 plays to Euripides, though we have only 18 now. He was known for changing the content of the plays by including strong female characters and intelligent slaves. Sophocles (about 496 BCE) wrote Antigone and Oedipus. He was attributed with 127 plays, though only 7 have survived. He was very well known because he won so many of the competitions that he entered.
There is an interesting timeline of Ancient Greek Theater published here http://www.greektheatre.gr/timeline.html if you like that sort of thing or if it helps you contextualize.