In 2-4 player matches on Mario Kart 64, Mini Bomb Karts are deployed all around the track, ready to explode anything that touches it. They drive left and right, and that's about it. Usually, they are stationed in the center of the track.
In Balloon Battle, 3-4 players, the first racer (or two if on 4 player) to lose is transformed into a Mini Bomb Kart in Mario Kart 64 and Mario Kart: Super Circuit. They can't pick up Items, but they can run into surviving players and destroy one of their balloons. However, it is a one time attack on MK64; you are just simply out, with an unmoving screen of the battle field. Mario Kart: Super Circuit lets you regenerate to damage opponents more than once.
Mini Bomb Karts are just a huge, black bomb with a pulsating yellow top and a lighted fuse set on 4, perfectly round wheels. Pretty simple.