Greek Drama
Apparently Aeschylus is the first dramatist to put more than one actor on the stage. Before this "drama" was more like our modern concerts or church sermon -- or both: contemporary church services continue the ancient "theatrical structure" -- a single speaker narrated the story, probably while strumming an instrument, and a group of singers (the chorus) accentuated the narration through singing.
The Chorus
Outside of the chapel, Broadway style musicals also maintain the ancient "choral" structure: music is interwoven into the drama to develop a deeper sense of emotional urgency, to express meaning emotionally rather than simply logically. Of course, Greek tragedies are cool and Broadway musicals suck, but that's a different subject. Most modern lyrical music (as in music with lyrics), whether it's pop, rock, metal, hip-hop, country, whatever, also continues the tradition of using a chorus to further develop verse. So thinking about the role of a chorus in modern music should help you understand the chorus in tragedy: it is essentially the same.
In Greek drama, the chorus generally comments on the actions or the antagonists, or sometimes they simply sing the praises of the relevant god or gods (see church sermon, above). The choral commentary generally represents the more objective, moral, and widely accepted view of the community; so, the characters represent how real people do act in real life, and the chorus generally tells the audience how those characters should act in real life.
We need to be careful with this explanation, however, because this isn't always the case. Keep in mind that Greek drama, especially tragedy, is intended to function on an emotional level -- and the choral music attempts to heighten emotion, the way a film soundtrack does today -- and emotion is by its very definition irrational, so the rational explanation in the previous paragraph isn't always accurate.
Also keep in mind that Greek drama is inherently religious, and again a religious, spiritual experience is irrational: it transcends reason. Also, most Greek religion probably makes little sense to us.
Finally, authors like Aeschylus were developing, inventing what we now think of as drama by blurring the lines between a traditional chorus -- which stands apart from the actor -- and what we now think of as drama: a group of actors interacting with each other. So, again like a modern musical, some Greek choruses also function as characters, like the "Leader" in the Oresteia.
Tragic Structure
In this class I'm far less interested in structure than in meaning, but here's an outline of the outline all Greek tragedies followed. It's interesting to see how it has translated itself into our modern drama. You will not need to know this on the test: