The theatrical tradition of the Western culture originated with the ancient Greeks. According to encyclopedia.com (linked below),
Tradition has it that at the Dionysia of 534 BC, during the reign of Pisistratus, the lead singer of the dithyramb, a man named Thespis , added to the chorus an actor with whom he carried on a dialogue, thus initiating the possibility of dramatic action. Thespis is credited with the invention of tragedy. Eventually, Aeschylus introduced a second actor to the drama and Sophocles a third, Sophocles' format being continued by Euripides , the last of the great classical Greek dramatists.
Greek drama was not like our plays today, however. Actors wore masks to represent their characters. There were no stages like the ones we're familiar with; instead, plays were performed in open-air amphitheaters. Many of those amphitheaters are still in use today.
See the eNotes articles linked below for information on style, themes, and other details of Greek drama.