The Cheshire cat listens to Alice and smiles at her, both of which encourage her. He also tries to explain to her the ways of Wonderland, though he does so in the typically bizarre way of this strange place. When she asks him which way she should go to get somewhere, he tells her that it doesn't matter: if she walks long enough, she will be "sure to" end up somewhere. He also informs Alice that everyone in Wonderland is "mad" or crazy and tells her she is mad too. This she feels to be untrue and she questions him:
"How do you know I’m mad?" said Alice.
"You must be," said the Cat, "or you wouldn’t have come here."
What the cat really means is that Alice simply needs to let go and let events happen in Wonderland because nothing here makes sense according to the logic she understands.
Alice looks for this cat throughout the book, as he can disappear and reappear at will. At the end of the book, he reveals the absurdity of the beheadings—when he is supposed to be beheaded, he appears only as a head, leading to a debate over whether a creature who is only a head can be beheaded.
The cat's main significance is that he tries to communicate with Alice.