It keeps asking me for my password, even when I'm using other apps such as Mail.
2 Answers
Try a reset - hold the home and power buttons down until the apple logo appears.
I had a similar problem when I'd downloaded an app update, but lost wifi, and it got caught in a loop, and the reset seemed to resolve it.
If a reset like Jaydles suggested doesn't work, syncing it to a computer with iTunes that has been registered with your account will resolve the issue.
