Invincibility is a special ability that allows the fighter or enemy wearing it ignore all sources of damage. The only ways to make a fighter invincible is to use Sticorey (though Stomar is invincible once he starts to attack, and Marcstick can utilize evade to ignore 75% of all damage), while Azures are capable of making any enemy (including Big Blue and other Azures) invincible.
Invincibility manifests itself in the form of a transparent sphere surrounding the respective fighter/enemy. Other than that, there's really nothing else to say about how invincibility looks.
Once invincible, a slush fighter will not react to any form of attacks but can still be targeted. If a Drab attempts to shoot the player in question, the bullets will pass right through. Blue punches will appear to connect, but deal no actual damage. Should the fighter come into contact with a Cobalt, the fighter won't suffer any negative effects. Also, it should be noted that an invincible fighter will directly attack his target without suffering damage from any enemies standing in his way.
Once an Azure has made an enemy untouchable, they can still be targeted but will not glow red to recieve damage or react to the hit. This is especially problematic for new players who, still unfamiliar with Azures and more concerned about keeping Big Blue airborne than doing some street cleaning, will impulsively call on their fighters to attack Big Blue. Unfortunately, while fighters appear to attack him, they produce no change on Big Blue's altitude and speed whatsoever.