How long does the laptop last on battery? If it doesn't even last 30 seconds, that sounds like you need a new battery.
To determine where the problem most likely is, remove the battery physically from the laptop. Then run on A/C power only. That will tell you if the power adapter is OK.
I've done this before and it operates fine. Where is the best place to buy a battery at?
I'm not sure where you live but Batteries Plus is pretty decent. You might be able to find one online as well.
I've run into this at work with the Dells we use. The "Fn+F2" combination toggles charging off/on. Why anyone would want to turn that off is beyond me, but I have seen it many times before.
What brand of laptop is it?