There are four possible causes of the activation error:
- The activation server is unavailable. Rare, but it happens. Usually fixed within minutes.
- Your antivirus or firewall (or your router's firewall) is blocking access to the URL gs.apple.com on ports 80 and 443.
- Your phone is jailbroken.
- Your phone isn't jailbroken, but the computer you are connected to during the restore or update was used to hack or jailbreak some other iOS device, and its network database was corrupted in the process.
Perhaps this will provide you some clues as to how to proceed. If your iPhone has not been jailbroken, Apple has some troubleshooting suggestions here:
If your iPhone has been jailbroken, you're on your own. We can provide no assistance for jailbroken devices in these forums.
(part of the above information quoted with gratitude from Lawrence Finch)