Could be the outlet is switched off or mis-wired, could be the charger has failed, can be a failed cable (newer Macs, where that is detachable), or can be the MacBook Air itself.
Swapping some parts around is typical testing, trying to see whether and where the problem moves.
Checking whether the charger is showing an indicator on the MagSafe, too—usually shows as green initially and quickly shifts to amber when changing, for instance.
Checking for charger cable damage, too. Some of the cabling near the MagSafe connectors from this era can fray, for instance.
If the charger won't charge another Mac, yes, it's usually the charger.
You can usually get Apple to check this stuff at a local Apple Store, if you bring in all the pieces.