When you want to gain commitment from others. When you need to work through hard feelings, animosity, etc. The best decisions are made by collaboration. The objective of compromise is to find some expedient, mutually acceptable solution which partially satisfies both parties. It falls in the middle group between competing and accommodating.
Compromise gives up more than competing, but is less than accommodating.
When the goals are moderately important and not worth the use of more assertive modes. When people of equal status are equally committed. To reach temporary settlement on complex issues. To reach expedient solutions on important issues. As a back-up mode when competition or collaboration don't work.