Greek drama, although the foundation of modern western drama, differs radically from contemporary theater in that it was part of the religious are well as secular life of the Greek polis. The origin of Greek drama was in choral religious celebrations at festivals of Dionysus; individual actors were a fairly late addition. The greatest expense in staging a drama was paying for the hiring and training of what was originally a 50-person chorus.
For Athenian festivals, playwrights submitted written scripts for plays to a committee. The top plays were presented at the festival and funded through a mechanism known as a "liturgy", in which the Athenian officials selected individual rich citizens each to fund a play (it was actually consider a major, if expensive, honor). The rich citizen who funded the production of a play was called the choregus; it is a function somewhat similar to the modern title of "producer". The best way to think of the difference between a choregus and a chorus is simply that the choregus was the person who paid for the chorus.