The "Gainax Bounce" refers to breasts bouncing around gratuitously in anime, often in ways that seem to defy physics.
While it has been used by numerous animation studios, the Gainax Bounce was popularized by the wider spread of various anime created by the GAINAX Studio. "Gainax Bouncing" or simply "Gainaxing" is usually used to refer to especially exaggerated bouncing of big boobs, and some would argue that true Gainaxing involves each breast bouncing individually.
Bouncing breasts have become a common sight in modern anime, to the point where many people automatically associate it with anime in general without a second thought. Generally, the use of this concept is found in anime that is targeted at teen or adult male audiences. The more dramatic the movement, the more dramatic the bouncing becomes. The human eye is naturally drawn to movement and this is used to draw the attention of the audience, though often for a comedic effect. Bouncing breasts is one of the most common forms of fan service, right alongside the panty shot.
A female character may make a rather subtle movement and her breasts will suddenly shake, jiggle or bounce--sometimes realistically--but more often than not in a somewhat exaggerated fashion. The amount of bouncing ranges from being subtle to being dangerous to anyone in close vicinity of the breasts. Gainax Bouncing is usually present in shows with heavy fan service, or fan service-themed episodes or scenes (such as those taking place at a beach or hot springs). Bras (and clothing in general) usually appear to have no effect in stopping a pair of bouncing breasts.
Bouncing breasts are also depicted in manga but obviously don't come across to the reader quite as effectively.
Official Artwork, particularly such featured in anime magazines like Megami and Nyantype, sometimes feature characters in mid-bounce - similarly to how bouncing breasts are shown in manga. This is usually to promote shows that feature heavy amounts of fan service, but due to the nature of such magazines the idea isn't limited to such shows. Fan art, naturally, is limited only by the artist's imagination.
Figures also sometimes feature characters in mid-bounce, such figures may be considered "ero" (erotic/adult) figures though this largely depends on the amount of skin on show and/or the pose of the character.