When I press the power button, the laptop will not report any signs of turning on. The charger is plugged into the laptop....

Your laptop may be running too many programs at the same time which will cause it to overheat. Make sure it is running on a flat surface so the computer has room to circulate air to keep it cool. Your fan may also be dirty and you can clean it or you can replace it using the replacement guide for the device. You can find a replacement guide here.