In the US, the Tooth Fairy legend typically involves children leaving a tooth under their pillow in exchange for money from the Tooth Fairy. But the Tooth Fairy hasn't always worked this way. In the Middle Ages in England, the tradition was to bury baby teeth that fell out, and around the world, many different customs have been followed, from throwing your tooth on the roof to wrapping it in a piece of bread and feeding it to an animal.
According to Wikipedia, the tradition of leaving your tooth under a pillow for The Tooth Fairy is believed to have originated from a French fairy tale. In the tale, a mouse changed into a fairy to defeat an evil king by hiding under his pillow and knocking out all his teeth. And in many parts of the world, including our neighboring Mexico, many children still hide their teeth for Raton De Los Dientes, or Tooth Mouse.
In the United States, a play called The Tooth Fairy was published in 1927, and an episode of the "The Little Rascals" from 1938 helped popularize the Tooth Fairy legend. The episode included a plan in which some of the kids agreed to pull their teeth out to make money from the Tooth Fairy. Interestingly, Wikipedia says the fairy's first major appearance as an illustration in popular culture in the United States came in the Peanuts comic strip. In a March 1961 strip, Frieda asks if the Tooth Fairy's prices are set by the American Dental Society.
A great book to learn more about tooth traditions around the world is "Throw Your Tooth on the Roof: Tooth Traditions from Around the World" by Selby Beeler.