Quick example: Brainwave readings (with an EEG machine) have shown that brainwaves of 2Hz occur when an individual is asleep. Scientists can create special audio recordings, which contain an encoded 2Hz frequency. While listening to such audio, the brain naturally follows (entrains) toward that 2Hz frequency, lowering its brainwave patterns. That's "brainwave entrainment." The net result is that when listening to such an audio recording, you (the listener) will naturally fall asleep.
Both brainwave entrainment and neurofeedback deal with brainwaves, but the similarity stops there. Entrainment pushes your whole brain into a pre-determined state, while neurofeedback teaches you how to move specific parts of your brain on your own. It is the differeence between forcing the brain into a given position, and skills building so you can move it there yourself.