Greek theater has made a big impact on Western theater. Actually, I have found 4 main reasons as to why.
First, the dialogues/ text play a huge part. Aristotle stated that the sort of language required of a tragedy was heightened language/ verse. The Western theatrical tradition (created by Shakespeare) owes a huge debt to the Greeks on this requirement of verse in drama.
Next, there'es the audience, obviously. This one isn't a big surprise. People can have fun performing with their friends, of course, though it isn't really a performance without an accurate audience. Greek stadiums have impacted us all, more specifically, Western theater.
Thirdly, the actors made an influence. Greek theater began with the idea that the performance was a group event whose players were known as the chorus, and their job was simply to narrate the story. Over time, first one actor emerged as the protagonist to speak solo lines, and then more "characters" stepped forward. These characters began to engage in conversation, or "dialogue," to enact rather than narrate the story. Thus, the idea that the actors don't simply tell a story but inhabit the characters and speak dialogue is an invention of the Greek theater. Though the chorus remained a part of Greek theater, the course of Western theater was forever changed.
Finally, he scenery comes into play. In the Greek theater. The actors made their entrances and exits from a building called a "skene," a term that gave rise to the Western concept of scene or scenery. By the time of Sophocles, there were actual painted backdrops to enhance the unchanging environment provided by the skene for each performance. The entrance of gods was staged by the effect of lowering the actor from the top of the skene, so that he flew above the stage. These simple devices are still employed today, and continue to be tested and developed, as the producers of the Broadway show "Spider Man" can attest. (Yes, it is spelled "skene", not scene.)
Once again, I hope I helped :)
