Collect all ten snow boys in any stage. The stage you collected all ten will unlock Boss Battle mode for the Boss of that stage.
Collecting Snow Boys
To find Snow Boys look for a place where a small amount of snow is falling and use the Blizzard attack. This will reveal Snow Boys. There are ten in every stage.
Hard mode
Successfully complete the game and view the ending sequence. You will start back at stage 1 with all the snowflakes and lives you had when you completed the game, except that minor enemies like snowmen with blue caps will shoot a fireball at you immediately after they are killed, making the game much more difficult.