Many people have reported issues with Windows Update in Windows 10. Check first that you've upgraded to the Windows 10 Fall update (see above, number 2). If you're still getting problems, download and run the Windows Update Troubleshooter, then reboot and try to update again.
If the problems remain, you might need to get a bit more stuck in. First, check that System Restore is configured (see below, number 7) and create a restore point. With this done, use Win+x and select Command Prompt (Admin), then type 'net stop wuauserv' (without the quotes) and hit Enter, followed by 'net stop bits' and Enter. You should see confirmations that each service was either stopped or wasn't running. Next, open Explorer, navigate to C:\Windows\SoftwareDistribution and delete its contents including any sub-folders. Now reboot open Windows Update, and click Check for updates.