That is a very interesting question and it happens to be on the fore-front of modern Astrophysics and General Relativity.
If two bodies (stars, black holes, anything) get close enough to be gravitationally bound they will develop some sort of orbit in which the two bodies circle each other.
Einstein's theory of General Relativity predicts that when two very massive objects(like Black Holes and Neutron Stars) are caught in an orbit like this they will generate Gravitational Waves (Ripples in Space-Time). These Gravitational Waves carry energy away from the orbiting bodies which means that with each successive revolution they get closer to each other. Eventually the orbit gets so small that the two bodies merge.
The merger of two black holes entails a lot of complicated General Relativity and some non-intuitive physics, but theorists use large supercomputers to simulate such events. There are two primary results to these mergers of which we can be sure of:
- A single, more massive black hole
- An extremely powerful burst of gravitational waves
There are currently several projects which are attempting to directly detect the disturbances in space time caused by gravitational waves. One is called LIGO, and they primarily expect to "see" the results of neutron star and black hole binary mergers. Another is called NANOGrav and the IPTA, and they expect to "see" the final stages and final mergers of pairs of supermassive black holes in the centers of colliding galaxies.
There is also quite a bit of evidence that black holes merge in distant galaxies from electromagnetic observations (light), from observations of jet streams caused by the moving black holes, to periodicities in the observed brightness of some of these systems, to simulations of galaxy formation and merger histories.
This page was last updated October 2, 2015.