its uses are too many to be covered by one site. the idea is to look at the duality between functions on a group such as R, and functions on the dual group, in this case the circle.
It has hugely many uses including generalizations to analysis of sheaves on abelian varieties and their dual varieties.why not just start searching and reading the sites you find? or go to the "library" (a building at your school where knowledge is stored in print in "books").