The Eject Button is just in front of the joystick. When your 'Mech is highly damaged and/or your 'Mech is completely out of weaponry or ammo, this button will flash. If you are in Mid or Advanced Pilot modes, you can hit this button to eject from the 'Mech and be transported into a new 'Mech.

This button is also known as the "Bonus Points" button by the Veterans. The reason for this misleading name is that every time you hit the Eject Button, you will receive negative points in your score (amount determined by mission type) and therefore allows the other vets to be that much higher in score than the one(s) who use that button.