They're a row of elements on the periodic table (lanthanides) which have a broad range of industrial uses (see the wiki). It is a misnomer, as many of them aren't particularly rare.

http://en.wikipedia.org/wiki/Lanthanide

http://en.wikipedia.org/wiki/Rare_earth_element

They occur all over the world. The US was once the leading producer, but (IMHO unreasonable) regulation forced its mines to close (they were in California). Currently China produces virtually all of them (97%-99%). There's a geopolitical stink about their protectionism: they're restricting exports, which (to my understanding) means they're intervening in favor of Chinese rare-earth consuming manufacturers (e.g. electronics industries). There's also human issues, in that the mines are a major pollution source and health hazard, and many (NYT estimates half) are operated illegally by crime syndicates.

http://pubs.usgs.gov/fs/2002/fs087-02/

http://www.reuters.com/article/2011/03/22/us-china-rareearth...

http://www.reuters.com/article/2011/07/14/us-china-rareearth...

http://www.economist.com/blogs/babbage/2010/09/rare-earth_me...

http://www.nytimes.com/2010/12/30/business/global/30smuggle....

Rare Earths are deposits of exotic minerals used in nearly every advanced technology today (wind mills to MRI's). California used to produce the entire world supply until China undercut the price. As demand boomed (prices going way up) and China started using them as a trade weapon so companies started looking for sources in more stable countries.