Within their division (the NFC South), the Atlanta Falcons are the Saints' biggest rival. They don't have a major ongoing rivalry with any one team outside their division unless you count the Chicago Bears, who beat the Saints in the NFC Championship game and kept them from making the Super Bowl after the 2006 season.
The New Orleans Saints, currently and pre-NFC South days. It's always been the Saints at #1 for rivals.
Broncos & Chargers; Redskins & Cowboys
I believe the Oakland Raiders are the Kansas City Chiefs biggest rival. The Chiefs lead the all time series 59-51-2 as of the 2013 season and lead 2-1 in playoff matches.
Kansas City Chiefs cheerleader, Susie Derouchey
Kansas City, Missouri.
Arrowhead Stadium, home of the Kansas City Chiefs, is located at One Arrowhead Drive, Kansas City, Missouri, Jackson County
The Chiefs name was derived from a poll held in the Kansas City area.
Josh Martin is number 95 on the Kansas City Chiefs.
Junior Hemingway is number 88 on the Kansas City Chiefs.
Justin Houston is number 50 on the Kansas City Chiefs.
Kyle Williams is number 19 on the Kansas City Chiefs.
Marcus Cooper is number 31 on the Kansas City Chiefs.
Mike DeVito is number 70 on the Kansas City Chiefs.
Nico Johnson is number 57 on the Kansas City Chiefs.