The general thrust of Super Mario Odyssey's story is similar to most Mario games.
Bowser abducted Princess Peach and Mario has to save her. This time he's assisted by a magic hat named Cappy, and flies in a hat-shaped spaceship called the Odyssey between different worlds. Well, Cappy isn't just a hat. He's a ghost hat that gives Mario magic powers; his "normal" form is a floating white top hat with eyes, and he seemed to simply possess Mario's red hat, replacing the trademark M on the front with those same eyes.