Space Bosses
Mirage Dragon
Keep shooting it in the mouth.
Hunter Phantron
This seems like a hybrid of the Phantron and the Plasma Hydra from the original Star Fox. Shoot at its body and try to avoid getting hit by its arms.
Land Bosses
Andross
It's similar to taking out a battleship core. After you destroy it, quickly destroy the "Andross cube" that appears. This will begin the final battle proper.
From here, Andross' battle has two phases on Normal, three phases on Hard, and four phases on Expert. The first phase is a large mask; shoot out the eyes. The second phase (Hard and Expert mode) will have a single eye as its weak point. The third phase (Expert mode only) will have two large hands you have to destroy, similar to Andross in Star Fox 64. The final phase, no matter the difficulty setting, will be the Andross cube you saw earlier. Quickly destroy it, and you beat the game!