I never know where to begin when talking about a dramaturg since the position can encompass so many different aspects. A dramaturg studies the scripts and researches the world of the play and shares this information with the cast and designers so they have a better understanding of the production.
For example, in Candide, there are a lot of historical events and ideas in both the book by Voltaire and the musical itself such as the Lisbon earthquake, Optimism, the Spanish Inquisition, and Jesuit ideology. During Voltaire's time, many readers would instantly understand these references. Today, we are farther removed from these concepts. To assist the actors during the rehearsal process, Jackie created a packet of information explaining many of the events and concepts as well as describing the historical milieu in which Voltaire wrote the book. In addition, she helped with pronunciation of certain words (like videlicet) and translation of some of the foreign words used in the lyrics. Information from her packet was later used by the Education Department to write their Study Guide.
The duties of a dramaturg can range farther than just research on a production. Many double as literary managers and read scripts and recommend plays for the theater to produce. Some are new work dramaturgs that are hired on a project by project basis to work with a playwright to shape a script and ready it for production. A dramaturg friend of mine is also the casting director at the theater she works at. It makes sense when you think about it because she so fully understands the world of the play, she can help in choosing the right type of actor for each role. It doesn't hurt that she has a natural talent for casting. Whatever their role, many dramaturgs spend years studying their specific type of dramaturgy. In recent years, universities have recognized the value of trained dramaturgs and have created Master's Programs in Dramaturgy.
There is also a formal organization dedicated to Literary Managers and Dramaturgs: http://www.lmda.org
The role of the dramaturg fascinates me. I think how much fun it would be to research all day without having to write a paper to turn in for a grade. While some of the research goes into program notes or study guides, the vast majority is shared only with the cast. But this wealth of information and little known facts that a dramaturg brings to a rehearsal process often gives the actors a new depth of understanding that manifests itself in more nuanced performances.