It's 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. We call it a family drama for the obvious reason that it concerns a family. Over the course of the play, we watch the Helmer family disintegrate as fast as Kool Aid in water.
We also dub it a tragedy, though it's a bit different than the Greek or Elizabethan versions. Ibsen's version of tragedy is all about the individual vs. a corrupt popular society. This is the opposite trajectory of a lot of previous tragedy. In Hamlet, for example, the good society has been thrown out of wack by the murderous, incestuous actions of Prince Hamlet's uncle. Hamlet must restore the kingdom to the lovely place of goodness that it once was.
In Ibsen's version of tragedy, society was never any good to begin with. A Doll's House, for example, shows Nora (and debatably all its characters) trapped in a society defined by restrictive gender roles. In order to become more than a doll, Nora must shatter the cornerstone that her entire society is based on: marriage. There you go: individual vs. corrupt popular society. In this tragedy, we don't get blood and death at the end; we get the death of a marriage and of the characters' old selves. Ibsen presents these things as the price of self-fulfillment.