The hookah-smoking caterpillar in "Alice's Adventures in Wonderland" is named The Caterpillar. This follows the trend Lewis Carroll used for many of his characters. The White Rabbit, the Queen of Hearts, the Cheshire Cat and most of the other characters in the story had no other names.
As seen in the full text of the novel available on Project Gutenberg, Alice first encounters The Caterpillar at the end of chapter four. In chapter five, titled "Advice from a Caterpillar," the two have a discussion about size while the Caterpillar smokes a hookah. This is the only time The Caterpillar appears in the novel.