Short for the expression "fan fiction", fanfics are stories written by fans based on a pre-existing property, but has no real impact on it's continuum. Often these are written as alternate story lines or sequels, but nothing is off limits. Just like with any fan made projects, fanfics are not limited to just anime, anything is fair game. The internet age brought on the largest surge of fanfics as it created a fast and easy way to both write and share works.
Similarly, "fan art" refers to art created by fans but based on a pre-existing preoperty. Any art created by an artist not attached to the property in question in fan art. Many professional artists create fan art based on properties of which they are fans, for example Koge Donbo has created a wide array of Harry Potter fan art.