If your computer simply isn't responding to any attempts to turn it on, you may be having power problems. Remember that there's a difference between not powering on and not booting, a computer with power problems won't light up or make any noises when the power button is pressed. If lights and fans are coming on but the operating system won't load, you may have a memory, hard drive, or even motherboard error instead.
As a first step, unplug the computer from power and remove any batteries, then press and hold the power button for 10 to 15 seconds. This will completely power cycle the computer, draining out any electricity that may be left lingering in its circuits. If you plug the computer back in and still have no luck, it's time to start troubleshooting the different stages of the journey between the wall and the computer.
Start with the surge protector to see if the computer behaves the same way if connected directly to the wall, or to another outlet that is known to be working normally. Look at the power brick if you've got a laptop. Most power bricks have two cords, one that runs from the outlet to the brick, and one that runs from the brick to the computer. If either of these cords can be detached from the brick, try again with a known good cord if you have one. If you've got a desktop, you'll usually just have one cable to check, the one that goes from the outlet to the back of the computer. If your laptop's cables and adapters are working normally, you've probably got a motherboard problem, and it's time to call support.
If you've got a desktop, your problem could be either with the motherboard or with the system's internal power supply. Again, a known-good power supply will tell you exactly which is the problem. If your computer will turn on but your battery won't charge, you've almost certainly got a bad battery. As always, try a known good battery in the computer (and, if you can, try the suspect battery in a laptop that is known to charge) and make sure it's not an issue with the contacts in the computer.