The usual things to try:
Restart -- not just turn off and on but hold the button for an extended period until it re-boots itself
Toggle wifi -- turn it off then back on. Forget the connection and then re-enter it.
Check your registration both via the device AND on your account at Amazon -- clear both and try again. (If you have more than one account, be sure you're registering to the account from which you want to get the books.)
With a device purchased from Amazon, one of those should work; if not you should contact them. They should be able to figure out the problem and, if it's a fault with a device still under warranty, they'll replace it.
So my next question is: Where did you get it?
If you did NOT buy it from Amazon, the problem could be that it's already registered to someone else's account and/or it's been reported as lost/stolen. That will prevent you from registering a device. And if you can't register the device, Amazon won't let you download content from the Zon to the device; even stuff you own.