Arizona Cardinals
When I grew up, the Cardinals were the punching bag of the NFC East. They were still in St. Louis and Neil Lomax took a pounding game after game. They last won the NFL championship in 1947. They've had occasional middling success and heartbreak playoff losses, but I often thank the football gods I didn't grow up a Cardinals fan. Incidentally, the team is not--strictly speaking--named after the cardinal bird. They are named after the color cardinal red, a dark, faded red color inherited by their original hand-me-down uniforms.
Older helmets
The Cardinals moved to Arizona in 1988. You'll notice that the cardinal logo is essentially the same. He looks a bit meaner in the updated version. I guess decades of losing will do that to a cardinal.
Joined American Professional Football Association (what is now the NFL) when it formed in 1920 as the Racine Cardinals, although the team's roots stretched back to 1898. Moved to Chicago proper in 1922. Moved to St. Louis in 1960. Moved to Phoenix in 1988. Renamed Arizona Cardinals in 1994.