The nicename is (usually) just a sanitized version of the username. It’s suppose to be ‘nice’ in the sense that it is the ‘nicename’ that is used as a slug, for example:
www.yoursite.com/author/my-nice-name/
will take you to the archive of the author with nicename ‘my-nice-name’.
The documentation simply describes it as
A string that contains a nicer looking name for the user.