Yes, we know, it's a crazy mixture – but Aeschylus was one wild and crazy guy. In fact, there are probably even more categories that Libation Bearers fits into, but to keep things simple, we'll stick to four.
The play is a tragedy because it focuses on the downfall of a great figure due to the inescapable power of fate and circumstances. Ah yes, but who is that great figure? Well, that's a good question. In one way, you could say that the downfall of Clytemnestra is tragic, because she has been riding on a cresting wave of power since the end of Agamemnon, the previous play. Fate and circumstance would come to her in the shape of Orestes, out to avenge his dad. That said, because we don't see too much of Clytemnestra in the play, we think the main tragic figure is Orestes, who bravely carries out his task of revenge, but ends up psychologically destroyed because of the nature of that task – killing his own mother.
From this account (the whole mom-killing bit), it should already be clear how this is a family drama. The play also falls into the psychological thriller and suspense genre because of the climactic psychological grudge-match between Orestes and Clytemnestra; even though he kills her, she scores the victory in terms of driving him out of his mind. Finally, you could also say that the play is a coming-of-age story, to the extent that Orestes is coming back to prove that he isn't a kid anymore by standing up to his father's killers. Unfortunately, Orestes's coming-of-age experience looks likely to leave him disturbed for the rest of his life.