The program is a specific hearing environment that has been put into the hearing aid. For instance, on my hearing aid, I have a general program, a directional program (turns off the side mics) and a music program (changes the frequency ranges to be more rich).
The number of channels is the number of frequency ranges your Audiologist can change. The more channels, the more customizable the hearing aid's performance is. The number of channels is not related to the number of programs (every program will have 20 channels).
Hope this helps!