Q: computer not restarting

Not a lot of info to go by (what was the old OS, what was the update, ...), but as a place to start, try rebooting with the command-R keys held at the chime.  If you can get into recovery, use disk utility to validate your boot drive.  Just select it and hit the key "repair disk".  Pay attention to any output and look at the bottom of the screen to see if the SMART status says verified

If the drive is good (no errors) and the smart status says "verified", you can close disk utility and select the reinstall.  If that still leaves you with a mac that won't boot.  The next advice you get will be to wipe your HD, reinstall the OS, then recovery your stuff from backup.  I really hope you have a backup if this is required.  For sure it's a trip to the apple store if you dont and they may or may not be able to recover your stuff.

Take a look at the suggestions in this link for possible solutions:

http://support.apple.com/kb/TS2570

Some more suggestions.

Startup - Gray, Blue or White screen at boot, w/spinner/progress bar

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Startup Issues - Resolve (2)