I'm going to assume that your data is safely backed up, and that you're not worried about retrieving your family photos, your almost-finished thesis, or your foolproof plan for world peace. That way, I can concentrate on saving the hardware. (Besides, data recovery is a whole other topic.)

Even if your laptop is plugged into AC power, a bad battery can keep electricity from getting through to the laptop proper. So the first thing to check is if it will run without the battery.

Unplug the laptop, remove the battery, plug the laptop back in, and keep your fingers crossed as you power up. If it boots, you have two choices. You can either buy a new battery, or resolve that your laptop will always need an AC outlet.

If the laptop still doesn't boot, make sure the AC adapter is working. if you know someone with the same model laptop, bring your laptop to their place and try it there.

If you're not so lucky, inspect the AC adapter for any signs of damage. Try another outlet. Listen closely for noise when you plug it in.