When you harmonize different opinions or ideas, you bring them to some kind of agreement. It's challenging to harmonize the different interests of a whole classroom full of children, but that's part of what a good teacher does.
When senators from different parties work together on a bill, they need to harmonize their different expectations and and priorities before they can come up with a good compromise. Another way to harmonize is to blend two singing voices or two instruments in a way that sounds pleasing. This is the oldest meaning of the word, which ultimately comes from the Greek harmonia, "concord of sounds" or "agreement."