Bug prevents windows from updating
On the next screen you see pick Troubleshoot, Advanced Options, Startup Settings and Restart, and you should then see the Safe Mode option appear: try running through the update process again if you can.
A quick search online will give you Safe Mode instructions for older versions of Windows.
In Windows 10 you can find the Windows Update page by launching the Settings app from the Start menu and clicking Update & Security – if there's something wrong and Windows knows what it is then you should find details here.
Windows 10 has actually streamlined the update process, so you should be seeing fewer errors.
Microsoft feels your pain: it knows the update process can cause problems every now and again, which is why it's developed a troubleshooter program specifically for it – search the old Control Panel for "troubleshooting", then select 'Fix problems with Windows Update' from the list on-screen.
Do you know why "have you tried turning it off and on again" is such an IT support cliché? There's no magic trick to this – it simply clears out your computer's temporary memory (including any stalled files or processes), and lets Windows start again from scratch with everything on the system.
If your updates are stuck in the background while you still have access to Windows, you can restart as normal; if they're stuck before or after the OS loads, you're going to have to hold down the power button and do a hard reset.