First, welcome to the forums.
Yes, your laptop will probably run without a battery but some won't.
Let's take things one step at a time. The first step is to remove ALL power. Remove the battery and disconnect the power cord. Now press the power button a few times to discharge what are called capacitors. Capacitors are sort of like little batteries. Reconnect the power cord and see what happens. You may or may not need to put the battery back in.
How old is this laptop and how long was the warranty?