Hi Christy and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
The folder with question mark is a serious issue. It indicates that either the system files are corrupted or the storage has failed. If this is just the system files, you can reinstall the OS X from the Mac OS X Utilities screen to fix the problem.
If you can get the other Mac, then you can connect both to recover your files and the process is called Firewire/Thunderbolt Target Disk Mode. See here --> https://support.apple.com/en-us/HT201462
If this is a Mid 2011 or later Macbook Air, then your Mac has the Thunderbolt port. If your Mom's Mac falls in any of the following below, then it also has the Thunderbolt port.
So you have to get the Thunderbolt cable to setup Target Disk Mode.
You can also buy Disk Warrior (powerful disk utility application) and try to repair your storage/memory or recover your files with it. If it is unable to fix the storage, then it will at least help you to recover the files that are readable and save them to an external drive. Get it from here : http://www.alsoft.com/diskwarrior/
Disk Warrior is costly. If you don't want to recover any files, then you don't have to buy Disk Warrior. It is not worth it if you don't have any important files/applications to recover.
If you have any questions or would like any additional information, please don't hesitate to ask.
Remember, this is not my final answer. If my solutions didn't work please don't rate me bad or poor. Instead please reply back to me and I will continue to help you until we fix the problem or you are satisfied.
You're most welcome and Good Night !
That cable is correct one. If you send me the link I will verify it.
The best way to know is by connecting both Macs in Target Disk Mode. Or boot into Utilities select Disk Utility and do you see anything in the left column
No, don't buy Thunderbolt to USB. It won't work. Only Thunderbolt to Thunderbolt will work.
Can you boot your Mac to OS X Utilities screen, then open Disk Utility and see if you can see the drive there ?