Try shining a bright flashlight at a sharp angle. If you faintly see the background or mouse movement then the display backlight has probably died.
Also try connecting to an external display. An HDMI TV works with the right cable. If your Air works with an external display then you know the problem is limited to the internal display.
Finally you can schedule a genius bar appointment for a professional diagnostic and repair quote.
Display problems can be fixed, but it may not always be cost effective to do so. Some repair providers will offer component level diagnostics/repairs, but many will only offer to replace the entire display assembly. Depending on how new and high-spec your Air repair may not be cost effective.