Compatibility has to do with how people or things get along. A married couple should have a lot of compatibility; people who always argue don't have any.
Compatibility is a huge issue in life. Every person has compatibility with some people and a lack of compatibility with others. Compatibility applies to things too; computer programs have compatibility with certain types of computers but not others. In all cases, compatibility has to do with getting along. People with compatibility also have chemistry — they are in harmony.