Fellowships "work" as additional specialized training in your field of interest. Most require a certain amount of GME, often in 1 certain field. As noted, you generally apply for them during your final year of residency (although there are differences). Usually things like in-training exam results, letters of rec, etc. play a heavy role in obtaining a fellowship.
Follow the links to FREIDA at: <a href="http://www.ama-assn.org" target="_blank">http://www.ama-assn.org</a>