Install or Reinstall Lion/Mountain Lion from Scratch
Be sure you backup your files to an external drive or second internal drive because the following procedure will remove everything from the hard drive.
Boot to the Recovery HD:
Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.
Erase the hard drive:
1. Select Disk Utility from the main menu and click on the Continue button.
2. After DU loads select your startup volume (usually Macintosh HD) from the
left side list. Click on the Erase tab in the DU main window.
3. Set the format type to Mac OS Extended (Journaled.) Optionally, click on
the Security button and set the Zero Data option to one-pass. Click on
the Erase button and wait until the process has completed.
4. Quit DU and return to the main menu.
Reinstall Lion: Select Reinstall Lion/Mountain Lion and click on the Install button.
Note: You will need an active Internet connection. I suggest using Ethernet if possible
because it is three times faster than wireless.
Yes, why? Are you only able to boot into safe mode? What do you have available with which to do a backup? All you need is an external drive large enough to hold the contents of your HDD. You can use Disk Utility to perform the backup.
After booting to the Recovery HD per the above and run Disk Utility just don't yet proceed further. Instead do this first:
Clone using Restore Option of Disk Utility
1. Select the destination volume from the left side list.
2. Click on the Restore tab in the DU main window.
3. Select the destination volume from the left side list and drag it to the Destination
4. Select the source volume from the left side list and drag it to the Source entry field.
5. Double-check you got it right, then click on the Restore button.
Destination means the external backup drive. Source means the internal startup drive.
After the clone is completed you can then move on to erasing the drive and reinstalling OS X.
If you were to use the external HDD to transfer files from your MBP to the Macbook Air, then we know that the HDD is formatted compatible with Macs.
Open Disk Utility on your MBP and select 'First Aid', and then select 'Verify Disk' and then 'Repair Disk' if necessary.
Then select the volume and run 'Verify Disk Permissions' and if necessary run 'Repair Disk Permissions'.
Post your results.
This a copy of mine: