MacBook not charging
As suggested, SMC reset should be your first port of call.

If that doesn't change anything, try removing the battery and see if the system recognises the change. (Status should change to "no battery"/battery icon with an x). If it changes, that pretty much means that the battery is toast.

I'm guessing that your battery has failed as this is the most likely cause of these symptoms. Cycle counts aren't a sure fire way of determining battery health, they can just die.

If your battery is less than 12 months old, you are entitled to a replacement from Apple.