Several people have come to this blog looking for the answer to the above question. Since the word dramaturg is in my blog's title and I do ostensibly identify as one (hello, my dusty little MFA), I figured I should help a googler or two out and answer the question. Or at least attempt to.
Dramaturgs tend to wear many hats at a theater. There's the production dramaturg, who does research on a specific play, helps with casting, and as one of my professors liked to stress, makes sure the play's story comes through loud and clear. A dramaturg situates herself in the world of the play; I guess you could say she acts as a gatekeeper or protector of the play, though that has weird connotations that I'd rather not connote. In addition to researching the play's story and world, the dramaturg also researches the play's performance history (ie what did other productions do with the play?) and response (criticism) to the play. Dramaturgs work directly with directors and help designers with research as well.
Dramaturgs also create ways to help the actors and audience into the play. This includes actor packets, which are full of research tidbits and pictures, and a program note and lobby display. Some dramaturgs do a presentation at the first rehearsal. My experience with this has been both incredibly unpleasant and all right.
Dramaturgs sometimes (most times, okay) fill the role of literary managers too. Literary managers deal with script/play submissions at a theater. Many theaters use some sort of database to track play submissions, LMs are responsible for maintaining that database.They read new plays or at least delegate the reading of new plays to script readers and respond to playwrights with either a yes or a no or a not this play, but do you have others? Literary managers help to choose the theater's season, help to arrange readings of new works, and work with literary agents and playwrights.
Some dramaturgs are also straddled with the role of education director. Dramaturgs who are also responsible for a theater's educational department arrange for school groups to come in to see the plays, create study guides for students, and lead talk backs and discussions.
Dramaturgs are also occasionally given the task of translating and adapting a work, which is my favorite aspect of dramaturgy.
I hope this description is helpful for all who are looking for an answer. If you want more information on dramaturgy and literary management, <a href="http://www.lmda.org">Literary Managers and Dramaturgs of the Americas</a> is a great resource.