Drama, Family Drama, Tragedy
Well, the play is definitely a drama, because, you know… it's a play, a piece of literature meant to be spoken by actors in front of a live audience. This particular drama centers on the trials and tribulations of the Loman family, making it a family drama.
Death of a Salesman is also in many ways a tragedy. You've got the basic ingredients here. A misguided person sets out to accomplish something that he thinks is the right thing, but ironically it is that very thing that causes pain and anguish to himself and everyone around him. Just add wide-sweeping themes that show just what's wrong with all of society, and voila…you've got a nice steaming dish of tragedy.
Of course, Death of a Salesman has a lot of differences from the ancient Greek version of the genre. There are no choruses in this play and the protagonist, Willy Loman, differs in several ways from a traditional tragic hero. The main difference is that he's not a king or mighty warrior of some kind—he's just a salesman. And an unsuccessful one at that. With Death of a Salesman, Arthur Miller set out to create what he called a "tragedy of the common man." He wanted to show that the sorrows of your average everyday guy were just as worthy of dramatization as those of kings.