What happens during an IOS update?

The updater basically switches the old iOS for the new one and installs any security updates. It will then take you through a setup process, just to verify your Apple ID details. Then you can continue to use you device like before, but with all the new features.

No files (including notes, pictures etc) or apps are deleted during the process, but occasionally old apps that haven't been updated for a long time will stop working (but this is very rare).

I still strongly recommend backing up your device to iCloud or a computer before carrying out the update just in case it fails for some reason.

Backing up an iOS device:

How to back up your iPhone, iPad, and iPod touch - Apple Support

Actually depends on how you update.

If you perform an update through iTunes on a computer then, yes, the old iOS version is removed and entirely replaced by a new version downloaded through itunes.

If the update is performed over the air (or on wifi), then only the changed files are replaced. .

Over the Air Updates are generally smaller than iTunes updates for this reason.

The device is completely wiped then the new iOS is placed in the device's EPROMs. There is no updating new files only as with an OS X update.