All My Sons qualifies as a drama because it's a play, a piece of literature that's never fully realized until it's put on stage in front of an audience. On the micro level, it's a family drama, for pretty self-explanatory reasons – it's a drama about a family. And it's a family drama in a similar way as Ibsen's Ghosts (Miller was a huge Ibsen fan). Both authors use the nuclear family to explore much bigger social issues. In All My Sons, Miller investigates middle-class nearsightedness through the story of Joe Keller.
When the conflicts of the Keller family spill out into the whole community, the play is elevated to another level all together. It becomes a tragedy. However, unlike Greek and Shakespearean tragedies, it's not about royalty or other big and powerful people. Joe Keller is a hard worker and a good father, but an abysmal member of society, whose downfall we get to witness. Arthur Miller would call the play a "tragedy of the common man." For more information on that, check out the link to his famous essay with the same title in our "Best of the Web" section.