While playing a level in multiplayer, the currently highest scoring player wears a golden crown. If another player overtakes them in score, the crown will transfer to them and they will say a special voice line.
"It didn't suit you anyway!" - Lemon
"Anomaly corrected!" - Aubergine
When the level is complete, the Crown-wearing player speaks the end-of-boss dialogue (if playing a boss level), and becomes the focus for the Score/Rank screen. The other players are free to move around and fire their weapons during this screen.
The Crown doesn't have any gameplay impact, it's just there for bragging rights and to motivate the players to try their best.
The easiest way to take the crown is to sabotage the current crown holder - either by killstealing them to break their combo, using Singularity/Plasma to create extra danger for them, bursting Tanks near them, etc. Obviously this is dickish and counter-productive, so make sure your fellow players agree that it's OK to do.