Q: Mac won't  start to login Screen

Do you have a good backup?

Try restarting from your Recovery volume. Follow the directions there, for your problem.

About the screens you see when your Mac starts up - Apple Support

Circle with a line (prohibitory sign) says, "I tried to start from the Volume you said, but some of the items there are missing or damaged".

Disk Utility ( First Aid ) sometimes fixes this, (or a Restart in Safe mode, which does that step on the way up). But sometimes it requires a re-install in-place.

By definition, a re-Install in place does not bulk-erase anything, but instead re-writes the individual files that make up Mac OS. However, it is Prudent to have a Trusted Backup before rewriting over 350,000 files that make up Mac OS.

Mac OS X- Gray screen or prohibitory sign appears during startup

You can try this:

Reinstall OS X Without Erasing Drive

 

       Please be sure you back up.

 

1. Restart the computer. Immediately after the chime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the the side list.
4. Click on the First Aid icon in the toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Reinstall OS X and click on the Continue button.

 

[Permission to use part or all of the above has been granted by Kappy, exclusively, to theratter.]

Everything you need to know about OS X Recovery

How to reinstall OS X on your Mac - Apple Support