In perpetuity means "forever." If you are banned from the library in perpetuity after that one time you stood on a table and played your trumpet, it means that you will never be allowed back.
Using the adverbial phrase in perpetuity is a formal way to say "forever and ever," or "indefinitely," or "until further notice." A truly sustainable form of energy will provide power in perpetuity, and the words of a brilliant, important poet will last in perpetuity. The root is the Latin word perpetuitatem, "uninterrupted duration or continuity,", and it's related to perpetual, or "never ending."