The Cheshire Cat is the cat of the Duchess. Alice meets it when she leaves the Duchess' house, and finds it in a tree. It constantly grins and can disappear and reappear whenever it likes. Sometimes it disappears and leaves its grin behind. The Cheshire Cat is the only character in Wonderland that listens to Alice. He also teaches Alice the 'rules' of Wonderland.
The White Rabbit is nervous and always in a hurry. Alice fascinated follows him, he gets things moving again whenever he appears during the story. He is a sort of a guide through Wonderland for the girl. It is the first Wonderland character Alice encounters.