Apple released the iOS 7.1.2 update earlier today and it seems for some people trying to install via Settings > General > Software Update, things are freezing before they finish. Over-the-air (OTA), on-device updates are typically the easiest, fastest way to update, but freezes are the absolute exception to that rule. So, how can you fix?
The answer seems to be a hard reboot. Just hold down the Home button and Sleep/Wake button, just like you would to take a screenshot, only don't let go. Keep holding them down until the screen turns off and then comes back on again with the Apple logo.
When it's done, go to Settings > General > About and make sure you're on iOS 7.1.2. If not, repeat the update process.
If your iOS 7.1.2 update is frozen, if you're progress bar isn't showing any progress, give the hard reboot a try and let me know if it works for you. And if you come across a better solution, let me know about that too!