In the USA and Canada, fanny is a synonym for ass, buttocks, et al, in polite company. In Britain and the rest of the English-speaking world, fanny is another word for vagina, pussy, quim, etc. See also: fanny batter.
Interestingly enough, Fanny/Fannie is also a feminine name, apparently a colloquial form of Frances or Francis.
Another word of similar differences in the USA and UK would be pants.