Slightly less short answer: You get a bigger black hole, and an awful lot of energy is released at the same time. The process of merging two black holes is interesting, as lots of things (like angular momentum) have to be conserved, and it actually turns out the full detailed description of what goes on is really, really complicated. Have a look at http://www.youtube.com/watch?v=4m-ZVsLf070 for a pretty animation, and http://www.youtube.com/watch?v=SOFmNNAE7gA for a pretty insane animation with a bit more detail hidden in it…
Good question. The black holes would attract each other and move together. Eventually, they would merge, and their “event horizons” would join together to form a single black hole. Gradually, the two black holes would join together to form a spherically shaped single black hole.
We have in fact discovered a few cases of black hole pairs. They are orbiting each other, but will eventually merge. Here’s a very pixellated image that we have observed:
Some scientists have made a computer simulation of a merger of two galaxies, each with a black hole at its centre. Eventually the black holes merge, with a single galaxy orbiting around it:
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