MacBook Air charger doesn't light up but is charging
It sounds like you have an issue with your SMC. Don't worry, it should be easy to fix, here is Apple's support document on it. SMC Reset

Since you have an Air, you will want to follow these steps.


  
  Shut down the Mac.
  Plug in the MagSafe or USB-C power adapter to a power source and to your Mac.
  Using the built-in keyboard, press Shift-Control-Option on the left side of the keyboard, then press the power button at the same time.
  Release all keys, then press the power button again to turn on your Mac
  


This can be a bit tricky to do, but try to follow the steps exactly as outlined

Good luck!