Q: Aperture Won't Start Up

Try starting Aperture with the Command + Option keys depressed.

Can you launch Aperture with the only the options key ⌥ held down? To test with a new, empty Aperture library?

If you can get Aperture to launch this way, go to the main menu bar and select the "Aperture" menu > About Aperture.  This panel will show you your serial number.

Does Aperture work on an empty library?

Did you yesterday upgrade software or run any software for the first time?

Where is your Aperture library located?

Final Cut Express

Look for a crash log of Aperture. The Final Cut reinstallation has been known to corrupt the Aperture installation.

  (Aperture 3 unexpectedly quits if Final Cut Express 4 is installed)

Does the error message show:

Dyld Error Message:

  Library not loaded: /Library/Frameworks/PluginManager.framework/Versions/B/PluginManager

  Referenced from: /Applications/Aperture.app/Contents/MacOS/Aperture

Then you need to reinstall MacOS X and not Aperture.

Install the system using the latest Combo Updater on top of your current system.

The latest Mountain Lion Combo Updater is here for downloading: 

OS X Mountain Lion Update v10.8.5 (Combo)

The easiest way to find crash log is the Console application. You can launch it from Applications > Utilities.

Click in the Toolbar - Show Log List.

In the log list in the sidebar disclose "User Diagnostic Reports" and "System Diagnostic Reports". Are their any Aperture logs?

Then select the most recent one and look for the Error Messages.

That is a pity.

If you have a full backup from before the time you reinstalled Final Cut, you could try to restore this dynamic library from your backup to the /Library folder:   /Library/Frameworks/PluginManager.framework/Versions/B/PluginManager

The less risky procedure would be to reinstall the system by booting into recovery.  The Combo installer does not reinstall all libraries, only what has changed after the installation of the base system.

To reinstall by booting into recovery see this knowledge base article:

OS X: About OS X Recovery  http://support.apple.com/kb/HT4718 

You're welcome.    Good Luck!