Russell puts the analogy as follows:
If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is an intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense. If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.
A similar analogy is used by Carl Sagan in The Demon-Haunted World: Science as a Candle in the Dark: "what's the difference between an invisible, incorporeal, floating dragon who spits heatless fire and no dragon at all?"
The purpose of the analogy is to put the burden of proof on the believer in God to demonstrate the existence of such a being, rather than the burden of proof being on the atheist to disprove the existence of God. A similar point is made by Anthony Flew in the chapter in God, Freedom and Immorality: A Critical Analysis entitled "The Presumption of Atheism".
Richard Dawkins has used the teapot analogy to respond to a form of agnosticism that seeks to treat atheism and theism as being equally likely due to the perception of there being no evidence to decide between them. But, says Dawkins, why pick out theism? If you cannot be an atheist because there isn't evidence to decide either way, why not hold the same agnostic position about the teapot? Surely, you must hold the teapot as being equally likely to exist and to not exist.