In a one-player game, when you reach N. Boss, loose both rounds right off then don't touch the controller. When the options screen appears, select "Vs. Battle" and player two will now control N. Boss.
Contra III: The Alien Wars
At the player select screen, press L, D, (D+L), L, and START in a fluid motion. You will hear a sound if it works.
See Game Ending:
At the title screen press LT, LT, and RT.
At the title screen press and hold (U+LT+RT) then press START.
On the first level, move quickly through the level without shooting or hitting anything (your score must be "0"). When you get to the "power unit", destroy it without hitting the gun turrets on either side (your score should be "2800"). Starting at the next stage, you will have a new super-weapon to select from your weapons menu.