The Fragile States Index is a yearly report published by the United States based Fund for Peace and the American magazine Foreign Policy since 2005.
http://fundforpeace.org/fsi/
In the literature the FSI is sometimes used as "Failed States Index", however there is only one index, as there is only one publication on this topic: the yearly report issued by Foreign Policy and the Fund For Peace. Literally, in my opinion, there is a slight difference: fragile states might easily become failed states. Practically, there is only one index based on a specific methodology: Fragile States Index.