What is the make and model of this laptop?
The power adapter is the primary power source for a laptop, the battery is considered to be a backup power source. When the power adapter is plugged in the battery can be charged, but the power adapter is providing the power to run the laptop. The laptop should be able to run with the adapter only with the battery removed.
If you are not getting any power with the adapter connected I would suggest using a volt meter to see what the output of the adapter is producing. If you look on the adapter you will find the output DC voltage.
You should also make sure that the receptacle you are plugged into is carrying line voltage.