A compromise is a way of settling differences by everybody making concessions. If you want to stay out until 10 and your friend wants to stay out until midnight, 11 is a good compromise.
Compromise comes from the Latin compromissum, which means "mutual promise." It can be a noun or a verb. If you compromise with your lab partner over how to analyze the experimental data, you find the middle ground between your two ideas. Compromise can also mean to erode or diminish. If you never repair your brakes, you will compromise the safety of the car. If you cheat, you compromise your integrity.