In Super Mario Kart, balloons rotate around the kart like Triple Green Shells, but from Mario Kart 64 and on, they are tied to the back of your vehicle. They pop if the player is hit by an Item or hit by a vehicle made invincible by a Star.
They also can vary in color depending on teams. If you are playing free-for-all, they are different colors. If you steal enough balloons with a Mushroom (only in Mario Kart: Double Dash!!, Mario Kart DS, and Mario Kart Wii), you can get an array of balloon colors.
In Mario Kart DS, at the beginning of the battle, you start with one and must inflate the rest. You have four spares. In Mario Kart Wii, you have three and play until they pop. Then you respawn again with three balloons. Losing all your balloons and having to respawn causes you and your team to lose a point.