In Bohr's derivation for his equation which defines the radii for various energy levels of an electron in an atom, he assumes (without any proof) that mvr = nh / 2*pi.

Louis de Broglie later explained this assumption as being derived from his theory of particle waves. Since an electron is so tiny, its wave nature is large enough to make a difference. So, de Broglie theorized, as an electron orbits the nucleus of an atom, in order for it to not lose energy it must be reinforcing itself as it makes each orbit. In other words, the electron is a standing wave.

Since the total distance around the orbit of an electron with radius r is 2*pi*r, for the electron to be a standing wave its wavelength must go evenly into 2*pi*r, so n*lambda = 2*pi*r. Since the de Broglie wavelength of an object is h / p, and p = mv (assuming the electron is moving at a speed much less than c), n * (h / mv) = 2*pi*r, and rearranging this equation yields nh / 2*pi = mvr.

For further reference:
Cutnell, John D. and Kenneth W. Johnson, Physics. New York: John Wiley & Sons, 1998 (4th edition),
or any good physics book.