If a Mac's internal drive gets full or "nearly filled", there may not be enough free space left for the OS to get booted and running.
What you're probably going to have to do is get booted and running from an EXTERNAL boot drive, then delete some stuff from the internal drive, after which there should be enough free space for the OS to boot.
There are things you haven't told us:
What YEAR is the MacBook Air?
What version of the OS is on it?
Do you have any kind of backup drive?
If so, what kind of backup do you keep?
What's your level of "Mac expertise"?
(If you don't keep a backup, now you know what folks keep them).
How I would handle the problem:
1. Get an EXTERNAL drive
2. Boot to internet recovery (Command-OPTION-R at boot)
3. Connect the external drive and use disk utility to erase it
BUT DO NOT ERASE THE INTERNAL DRIVE !!!!!
4. Use the OS installer to install a copy of the OS onto the EXTERNAL drive
5. Set up a basic account (just username and password, skip everything else) on the external drive.
Then...
I would open and start browsing through the INTERNAL drive, looking for things that can safely be deleted.
Such things might be mostly movies (if you've saved any), as these tend to take up a LOT of space. There could also be time machine "local snapshots" eating up space.
What you throw out is up to you, but BE CAREFUL.
You need to get to having about 10gb of free space, at least.
Then...
Power down, disconnect the external drive (for now), and try booting again from the internal drive.
If this seems beyond what you feel confident in doing, it might be time to arrange a visit to an Apple Store genius bar if you have one within reasonable distance...