Drama, Family Drama, Realism, Tragicomedy
All right, the first genre on our list, drama, is pretty easy to check off. It's a play, meant to be staged in a theater. No narrative points of view, just what people say and do. Get it? Got it? Good.
Next up, this is a family drama if there ever were one. Almost everyone in the play is related to the others in some screwy way or another, like the brother of someone's dead first wife, for example. The first and second wives, the relationships that are based on marriage, not blood, all add up to create a family in which everybody tries to figure out who they are based on their position within the group. It's a free-for-all, and kind of funny.
Which leads us to tragicomedy. The play is actually based on a play Chekhov wrote a few years earlier, which was a tragedy. In that version, called The Wood Demon, Uncle Vanya actually kills himself rather than trying and failing to murder Serebryakov. This failure puts the comedy on the tragi- for this play, and that means that sometimes you might be confused about whether you should laugh or cry, or both.
To wrap it up, Chekhov is one of the heavy-hitters of 19th-century Realism. That's a literary movement that tried to portray people as they really were. It tried to show how people really talked and how they really lived. The mix of classes (Telegin and Marina thrown in with Serebryakov and Astrov) is a big clue that this play is in the Realist genre.