Extra levels:

Press L, Up, R, Down at the title screen. A creature will appear in the
bottom right of the screen to confirm correct code entry. Select puzzle
mode, then choose the new "Another World" option. Alternatively, press A,
Up, B, Down at the title screen. 

Play as Bub:

Press Z + L + R + A at the game selection screen. 

Level skip:

Press Z + L + R + A during game play. 

Bonus characters:

Select "Puzzle Mode". Then, press Left(2), Up, Down, L, R, L, R, L + R
at the puzzle selection screen. Bub, Bob, and the Boss characters may be
selected.