The dissolution of a relationship means that it's broken up or ended. The dissolution of your band means you better get started on your solo album.
Dissolution comes from the Latin word dissolutio, meaning "a dissolving of something." Dissolution looks very similar to "dissolve," so to help you remember the meaning, think about what happens if you put paper in water — it breaks apart. A dissolution of a marriage is the same thing as divorce. Although it sounds like disillusion, if you try to use them interchangeably, your logic will fall apart.