If it's an official firmware update from the manufacturer, usually done from settings, you shouldn't have to backup, unless they state that you must. It's always wise to backup anyway before doing anything like this, in case the update goes wrong, and you have to factory reset or something. And it is normal for most devices to reboot into recovery, and the update file is automatically applied, then it reboots, optimizes apps etc. which might take a while.
Be warned, if the device is rooted or otherwise modified, manufacturer updates can frequently go wrong, ending up in a bricked device.
If an update notification appears as a browser pop-up, then that's definitely not official. It's some website trying to trick you into installing something, likely malicious.