Short answers, pulled from the story.
Fan art is artwork created by fans of a work of fiction or celebrity, depicting events, characters, or other aspects of that work. It is not created, commissioned, or endorsed by the original creators, which distinguishes it from officially licensed artwork.
The legal status of fan art in the United States is uncertain. Fan art based on existing characters could be classified as a derivative work under 17 U.S.C. section 106, making its display and distribution unlawful without permission. However, a fair use exemption under 17 U.S.C. section 107 may apply, and courts assess this on a case-by-case basis using a multi-factor test.
Courts evaluate the amount and substantiality of the original work appropriated, the transformative nature of the derivative work, whether the use was educational or non-commercial, and the economic effect on the copyright holder's ability to exploit their own derivative works. No single factor is alone decisive.
Bryan Konietzko noted that during Avatar (2005-2008), typical fan art was a crayon drawing sent in an envelope. By Korra (2012-2014), fans were taking his skewed phone screenshots, un-skewing them, separating character levels, animating blinking and talking, and producing GIFs, all in a short time.
The Hugo Award for Best Fan Artist has been given each year since 1967 to artists who create original science fiction or fantasy artwork appearing in fanzines, semiprozines, or at science fiction convention art shows. It recognizes original fan artwork rather than derivative works based on existing characters.
In science fiction fandom, fan art traditionally refers to original artwork related to science fiction or fantasy, created by fan artists for fanzines, semiprozines, and convention art shows. The more recent general usage refers to derivative artwork based on existing fictional characters or celebrities, and these two meanings are now sometimes confused.