You seem to have a dead battery, you can check the status by going to
Apple Menu > About this Mac > System Information > System Report (10.7 or later, similar but different on older OS)
Your battery could die for a lot of reasons, it's simply been used a lot, or it's been damaged, or used in environmental conditions that are outside of it's operating spec etc. Check this link to see what a normal cycle count should be so you can at least see if yours has died from old age if nothing else.
You don't say what model of Air it is - if it's an original one, it may have an expected cycle count of as little as 300 (and even then, to acheive that you would have needed to have used it correctly in terms of not storing it with a full or empty charge, regularly fully depleting it etc). Batteries are weird. I had one go on my first Macbook seemingly simply because I left it flat for a month.