About

Fairy Bounceis a meme revolving around the sprite animations of a character from an arcade puzzle game in which that character, a three-eyed almost entirely naked girl, is seen jiggling and bouncing her breasts in self-praise as a winning move.
The subsequent popularity of the animation on Internet, especially as an avatar tool, can be considered similar to the Tuzki character.

The Magical Drop game series and its characters

Magical Drop (マジカルドロップ), sometimes referred to by native Japanese users as MagiDro (マジドロ), is a series of puzzle games originally released in the arcade, and later primarily released for several platforms such as the Neo Geo, Super Famicom, Sega Saturn and Neo Geo Pocket Color developed by Data East. The games are notable for being extremely fast-paced.

(Source : Wikipedia)

The following video shows a bit of what its gameplay is about :

The characters are based on the 21 major arcana from the Tarot card game, redrawn in a more moe style.
One of them, named World (The World, 21st arcana), is depicted as a cute three-eyed girl almost entirely naked who wears some kind of floating towel hiding her private parts (shown below).

Although she is already present in Magical Drop 2, the sprites and animations that would be reused in the meme come from Magical Drop 3, first released in 1997.

One of its earliest transcriptions as separate sprites on Internet is from Game Sprite Archives, in January 2002.

On the web : Fairy Bounce

There is no archive of when exactly the modified versions from the gifs began appearing on the web.
However, it had been used in banner ads for hentai websites in 2001. The ads did change the color of the original character in order to avoid copyright. This is confirmed by the following picture marked as being last modified in June 5th 2002, and filled under "Adult ads" on Newgrounds.
Then, in August 2002 and still on Newgrounds, an user by the name of Adam_Z made a modified version of the red-head fairy. Finally, a Newgrounds thread featured the blue-haired one in October 2002.
It seems that the meme, in its early stages, was picked by Newgrounds users as a funny animation.

Unaware of the real name from the character of an obscure arcade game, a recurring filename for it has been assumed as Fairy Bounce, which is still the most frequent name used to refer to the gif animation nowadays :

– Google results for Fairy Bounce.
– Google results for Fairy_bounce

A series of forum threads, talking about the crazy phenomenon of Fairy Bounce gif animations, on the dslreports website started being made in the end of 2002, the first dating from December 28. Entitled Can someone make this into an avatar for me?, an user asked to have an alteration from the original gif animation, , being remade into an avatar.
The thread garnered 7 pages of comments and derivatives into several different colors.

On those forums alone, about 7 different threads between 2002 and 2004 would be made about that phenomenon, showing the animation as being a very popular avatar at that time. Moreover, several websites hosting collections are mentioned as well. One of them was http://pages.sbcglobal.net/darkfire/darkfire.html, a personal website that is now defunct and was meant as an avatar generator.

Another one, http://www.twilightguild.org/images/signatures/FairyDance.htm, is also down but a wayback machine archive shows that it was working at least from December 20 2002 to June 3 2003.

Another gif collection would be hosted at the following link in July 2004.

Spread

The animations have become popular as avatars throughtout the years. A few examples can be seen with the links below :

– Forum avatar.
– Forum signature.
– As a screensaver.
– As a way to add your own touch to a webpage (somewhat NSFW).
– Newground forum thread about it.
– As a funny animation.
– One YTMND instance from 2006.
– Collection from Punkz0rz.com.
– Deviant Art Flash animation=.

Urban Dictionary entry in 2004.
Several websites offering collection of various gif animations have also featured them.