A broken gameplay element (whether it be a car, stage, event, situation, or combination thereof) is so overpowering that it ruins gameplay by supplanting other gameplay elements.
A classic example is Radical One using the Juggling Strategy on Do The Snake Dance and The Garden of the King; he can constnatly rack up insane amounts of damage and thus Waste others quite easily. This type of unfair strategy is "broken" and action is taken (in this example, banning Radical One from the mentioned stages) to prevent it from impacting competitive play.
The term "broken" is often used when describing techniques, characters, attacks, etc. in many competitive games. While the term is prevalent among many games, it is often overused. Many players are quick to label strong techniques as "broken" when in reality there are consistent counters to those techniques that have not yet been discovered (although whether one can say a technique is not broken just because of undiscovered counter-techniques is fuzzy). If a broken technique is not dealt with, the game would simply degenerate into both players trying to be the first to make use of the broken element as no other element would prove effective.