Short answers, pulled from the story.
Fan art refers to artwork featuring aspects of a work of fiction created by a fan that was not created, commissioned, nor endorsed by the creators. These creations exist outside the official scope of the original work while still depicting its core elements.
A crayon drawing stuffed in an envelope arrived at the office of Bryan Konietzko during the run of Avatar: The Last Airbender between 2005 and 2008. That simple piece of paper represented a specific type of fan labor that was not created, commissioned, nor endorsed by the creators of the work.
Broad availability of digital image processing tools greatly increased the scope of these creations starting when Bryan Konietzko began working on Korra between 2012 and 2014. Text-to-image generators now expand the potential reach even further compared to physical mail delivery.
The Hugo Award for Best Fan Artist has been given each year since 1967 to individuals who create such works within science fiction fandom. This traditional definition describes original artwork rather than derivative pieces found in semiprozines or fanzines.
United States copyright law controls the right to reproduce and display pieces of artwork under section 106 of title 17 U.S.C. A fan using settings and characters from a previously created work could be considered a derivative work requiring permission from the owner of the original work.