MacBook Air password recovery

Short of...

• remembering what the password is either directly or using any password hint that might have been set...
• of using your Apple ID and password to reset the password if that feature was previously enabled...
• of brute-forcing a too-short password...

there's intentionally no way to do what you want here.

Only a password hash is stored, and not an actual password. There's deliberately no cleartext password stored. So there's no saved password to recover.

Another administrative user can reset the password. There are also other ways to reset a forgotten admin password posted around the 'net. Apple has some password-reset information posted, as well.

That password reset means the keychain contents are inaccessible, and potentially more depending on the system configuration. Worst case, the system will need to be wiped and reinstalled.

Definitely do the procedure listed in MrHoffman's reply first.

If that doesn't work, however, the fact that you are running on an older system means you should most likely be able to do the following process to make a new Admin user if you didn't have FileVault (encryption) enabled. With a new Admin user it's easy to reset passwords (but as pointed out, you'll still need to reset your Keychain and lose all your saved passwords):

1) Boot up the mac holding COMMAND S. This will boot you into “single user” (text only) mode.

2) Type the following commands:

mount -uw /
rm /var/db/.applesetupdone
shutdown -r now

If you don't get any error you will have removed the hidden .applesetupdone flag file and this will kick your Mac into the out-of-the-box new Admin account setup!

You well see your “Welcome” mac user screen. No worries! This will not affect anything you have on the Mac. Your regular user account, data, apps etc., will be just fine. This process is just coaxing the Mac to generate a new Admin user. Just give it a name like "Test" or whatever.

You can log into that "Test" account and reset the password for your other real user account(s) in System Preferences > Users & Groups.

User Password Reset

User Password Reset (2)