It's a bizarre practice, but considered normal among many people I know in the U.S. I remember my mother keeping our teeth we lost as kids in a jar or something in her room.
When did this become normal? When-ish did it start? What sort of cultural phenomena such as this occurs else where?
This is a very old tradition and North America inherited it, with modifications, based on a European prototype. This this article for example. There are also some excellent articles dealing with this topic in Peter Narváez, ed., The Good People: New Fairylore Essays (Lexington: University of Kentucky, 1997). I'm on the run and don't have time to summarize for you, but the link should give you a good start.