MacBook Air stuck on log in

Reinstall Sierra or El Capitan Without Erasing Drive

Please be sure you back up.

1. Restart the computer. Immediately after the chime hold down the Command and R keys until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the side list.
4. Click on the First Aid icon in the toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Reinstall OS X and click on the Continue button.

Also, see this User Tip: Basic Steps for Upgrading OS X.

Reinstall Lion To Yosemite Without Erasing the Drive

Be sure to backup before proceeding.

1. Restart the computer and after the chime hold down the Command and R keys until the Utility Menu appears.
2. Choose Disk Utility from the Utility Menu and click on the Continue button.
3. After Disk Utility loads select the indented (usually, Macintosh HD) entry from the side list. Click on the First Aid tab, then click on the Repair Disk button. If Disk Utility reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button.
4. When the process finishes, quit Disk Utility and return to the Utility Menu.
5. Select Reinstall OS X and click on the Continue button.

That means you are trying to install Sierra on a volume that already has Sierra installed. Which of the procedures are you using? You should only use one or the other depending upon what is now installed on the computer. And, you need to follow the instructions carefully. The error should only occur when you are using an installer that is for an older version of OS X.

If you can't, you can't. Neither procedure will remove your data. The backup is precautionary.