What is a matter (de Broglie) wave?
A matter wave is the wave produced by particles.
wavelength = Planck's constant / momentum
Since light was shown to have a wave-particle duality by Einstein; Louis de Broglie suggested that matter should have a dual nature as well. He proposed that since light which is mostly wave has particle properties, then matter which is mostly particle should have wave properties.
The De Broglie hypothesis was not accepted at first because de Broglie had no empirical evidence to support his claim. His whole idea was based on mathematics and a hunch that nature was symmetrical. Einstein came to his defense and in 1927 Davisson and Germer confirmed the De Broglie hypothesis with empirical evidence.
De Broglie's hypothesis and Bohr's idea of energy levels would lead to Schodinger's equation and quantum mechanics.