Can be one of two things, faulty AC adapter or faulty battery. How was your battery before this problem began to occur? If battery life was under half an hour per full charge, it is time to replace it. The chances of a bad AC adapter is smaller than the chances of a bad battery.
In your case, the laptop will not power on with the AC adapter without the battery (or with the battery). It might very well be that both of them are bad.