What Happens If Two Black Holes Meet?

Difficult to say exactly. First of all such an event would be very hard to set up.

Next, the actual thing that a black hole really is is a dimensionless point, the singularity, at the center of the area that we call a black hole. The "black hole" is actually the event Horizon. The area between the event Horizon and the point black hole is a sort of no mans land.

Anyway, so if you could get two black holes of about the same size and the right rotational speed (there could be frame dragging effects) and aimed them exactly the right way they would collide and form a bigger black hole.

But if your aim was off or the rotation was wrong we really do not know what would happen. If the singularity of each black hole entered the event horizions of each other the only way they could get away from each other would be to travel faster then light, a very difficult thing to do.