Boss Mode gives you access to all the Bosses you have discovered in Campaign Mode. This allows you to practice against any particular Boss.
You can switch between Big Boss (Wave Bosses) and Mini Bosses mode.
In Big Boss mode, you can select Easy, Medium or Hard. You must beat the Boss at Easy before moving onto Medium, and Medium before Hard.
Once you beat a Boss, you have a chance to win a Crystal. If you beat that boss the required number of times (displayed next to the crystal) then you win a Light. Once you get all three Lights for that boss, the Boss is weakened by 5% within games, making it easier to defeat.