The new software iOS 8.1.3 won't update
Try updating through iTunes. This is useful for those times when there isn't enough free memory available for the update to install on the device itself.

The sequence is to back up the phone and then apply the update.

If there is an error, you will see it in the iTunes update error logs. You can also restore the device to factory settings (via iTunes or via the reset options in the settings app - erase all content and settings) and then update if the cause of the error is some corruption of the apps and data on the phone.