two logings

What is the exact model of your Mac and version of macOS used? If you cannot log in, then you can get the exact model by entering the system serial number here:

Check Your Service and Support Coverage - Apple Support

Is Filevault enabled?

Have you tried booting into Safe Mode?

Are you saying you get a login screen, enter the password and the system continues booting until you get to a second login screen? Then at this second login screen you are having issues?

If so, then try using another keyboard. I've seen some laptops where the keyboard may work before logging in, but fail after logging in.

Make sure the Caps Lock was not accidentally activated for the second login.

Do you have more than one admin user account on this Mac?

If you have a MBAir 10,1, then it is an M1 Apple Silicon Mac. I'm going to ask the moderators to move this thread to the main MacBook Air forum where current models are discussed so that more people are likely to see your thread since you posted in the MacBook 2006-2010 forum (no air or pro in the forum name, plus it deals with 2006-2010 obsolete models).

For an Apple Silicon Mac, there is only one way to have two logins available...that is when Filevault is enabled as it adds an extra password to the system. Your Filevault password may be (and probably is) different from the macOS login password. Try using all of the passwords you have ever utilized on this laptop for the second password prompt.

You can try putting the laptop into Target Disk Mode where it will allow you to connect it externally to another Mac in an attempt to access the data on the MBAir's SSD. I'm guessing you will have the exact same issue, but it is worth a try. This requires access to another Mac running macOS 10.13+, but it would be best if the host Mac is running 11.x+ as it should make it easier to access the "Data" volume if the SSD is unlocked & authenticated.

You can also try booting into Recovery Mode to see whether you can access the utilities within Recovery Mode. If you can boot into Recovery Mode, then launch Disk Utility to see whether the "Data" volume is mounted. If not, then select the "Data" volume and click the "Mount" button. Double check to see if Disk Utility shows a path for the "Data" volume to indicate the volume was successfully mounted. If it was mounted, then you may be able to use the command line within the Terminal app to transfer your files to external media, but this does require knowledge of the command line.

If you can successfully reach the recovery mode screen, then you should be able to boot the laptop into Safe Mode or access the data using Target Disk Mode. I doubt if any of these suggestions will work though, but they are your only options if resetting the password of the macOS user account does not work.

You can take the laptop to an Apple Store or an Apple Authorized Service Provider to confirm the problem and that you've performed all the steps properly.

Otherwise your data is lost if you do not have any backups. In that case your only option is to perform a firmware "Restore" which will reset the entire laptop, erasing the internal SSD, and pushing a clean copy of macOS onto the internal SSD. This process will destroy all data on the internal SSD. This process also requires access to another Mac running macOS 12.4+ (at least the last time I check, unfortunately Apple keeps increasing the OS requirements for the host Mac).

If you don't have any backed, then in the future make sure to always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data. macOS includes the Time Machine backup software for free with macOS.