Shut down because of an error

After it restarted, did you see a dialog box resembling this one:

If so, next time it occurs click Report...

Before you send it to Apple, copy the text of the report. Paste it in a reply. Remove or obscure any personal information, should it appear.

If problems continues to occur please determine if they also occur in "Safe Mode":

• Safe Mode or "Safe Boot" is a troubleshooting mode that bypasses all third party system extensions and loads only required system components. Read about it: Starting up in Safe Mode
• Starting your Mac in Safe Mode will take longer than usual, graphics will not render smoothly, audio is disabled on some Macs, and some programs (iTunes for example) may not work at all.
• To end Safe Mode restart your Mac normally. Shutdown will take longer as well.

Perform anSMC reset.

Read all the steps. If the MacBook Air never gets beyond the login screen before it spontaneously shuts down you won't be able to perform the "Before Resetting the SMC" steps, so just skip to the next section.

Before Resetting the SMC

Try each of the following steps in this order before you reset the SMC. Test the issue after completing each troubleshooting step to determine if the issue still occurs.

1. Press Command + Option + Escape to force quit any application that is not responding.
2. Put your Mac to sleep by choosing the Apple () menu from the upper-left menu bar and then choosing Sleep. Wake the computer after it has gone to sleep.
3. Restart your Mac by choosing the Apple () menu from the upper-left menu bar and then choosing Restart.
4. Shut down your Mac by choosing the Apple () menu from the upper-left menu bar and then choosing Shut Down.

Resetting the SMC on portables with a battery you should not remove on your own

1. Shut down the computer.
2. Plug in the MagSafe power adapter to a power source, connecting it to the Mac if it's not already connected.
3. On the built-in keyboard, press the (left side) Shift-Control-Option keys and the power button at the same time.
4. Release all the keys and the power button at the same time.
5. Press the power button to turn on the computer.

Determine if the Mac starts and allows you to log in.

Not to over ride John. It might be worth trying to reset PRAM.

http://support.apple.com/kb/HT1379

Do you hear any odd nosies coming from your unit (loud fan, beeping)?

You can also try booting in verbose mode (command + v on startup) to see if it crashes in the same place every time.

Also try booting to recovery and seeing if you can open disk utility and running verify/repair disk on your HD.

Resetting the NVRAM won't hurt.

If your MacBook Air has either OS X "Lion" or "Mountain Lion" installed attempt to boot OS X Recovery: Immediately after pressing the power on button, hold both the ⌘ and r keys (two fingers) until the Apple icon or "spinning globe" appears. When the Mac OS X Utilities screen appears as in the screenshot below, select Reinstall Mac OS X and proceed with the Continue button:

You will need an active wireless Internet connection to download OS X from Apple.

Reinstalling OS X does not erase your User accounts or data, though it is possible that whatever is causing the Mac to not start may have resulted in some data corruption that can be fixed by restoring from a backup.

Assuming OS X is successfully reinstalled, attempt to start in Safe Mode again, followed by starting it normally.

If you have a version of OS X that predated Lion, or if neither the Apple logo or "spinning globe" appear after attempting to use OS X Recovery, only Apple will be able to fix whatever is wrong. In that case, schedule a Genius Bar reservation:

http://www.apple.com/retail/geniusbar/

A Time Machine backup would have been stored on an Apple Time Capsule or external hard disk drive. If you did not use Time Machine, there is no backup to choose.

The internal drive is malfunctioning, or you have some other hardware fault.

Make a "Genius" appointment at an Apple Store, or go to another authorized service provider.

Back up all data on the internal drive(s) before you hand over your computer to anyone. There are ways to back up a computer that isn't fully functional — ask if you need guidance.

If privacy is a concern, erase the data partition(s) with the option to write zeros* (do this only if you have at least two complete, independent backups, and you know how to restore to an empty drive from any of them.) Don’t erase the recovery partition, if present.

Keeping your confidential data secure during hardware repair

*An SSD doesn't need to be zeroed.

You ought to be able to purchase it from Apple for a nominal fee. They will require your Mac's serial number.