A computer that seems dead may actually be running with no picture, or it may have no power at all. The first step is to separate those two possibilities, because the fixes are completely different.
Listen and look for signs of life. Fans, lights and hard-drive sounds mean the machine is getting power and the problem is likely the display or a stuck low-power state. Total silence points to power supply, battery, cable or button.
Laptops can get stuck in a sleep or hibernation state where the power button does nothing. A forced reset, usually a long press of ten to thirty seconds, drains residual power and lets the system start fresh.
Power adapters and cables fail more often than the computer itself. A worn cable may carry enough power to show a charge light but not enough to run the machine, which is why trying another adapter is such a useful test.