The first boss will appear in Stage 1-2 (Ghosts). It will alternate between two types of attacks. When its ship is pink, it will spray the screen with bullets. Do your best to avoid them, whilst shooting the boss. When the boss turns blue, it will shoot a powerful beam. As soon as you see it change to blue, move to the far right of screen. It cannot hit you there. Once it changes back to pink, resume shooting and avoiding its bullets. Continue this pattern until the boss is destroyed.
Upon killing the boss, many power-ups will appear, including the Beam-Shot. Simply move your ship onto the largest power-up to unlock this achievement.