Permalinks are used to make album urls nicer by associating an specific name to an album. When an album has a permalink defined, it can be reached by it´s permalink name so the id of the album is not required anymore in the url.
Note that permalinks must be unique per album. Also in the permalink history, it cannot have the same permalink defined more than once.
Permalinks: list available permalinks and the albums it points to.
Permalink History: When a permalink is deleted, you can save it in the permalink history, so that external links to Piwigo pages still work. In the permalink history box you can see the date when the permalink has been deleted, the last time it was used and the number of times this permalink has been used.
URLs can also become nicer by changing question_mark_in_urls and php_extension_in_urls from configuration file but it depends on server specific configuration and may not be available.