If Nothing Happens
then one of the following things may be the source of the problem:
1. Your computer may be unplugged. I know it sounds silly, but stranger things have happened. Pets, children, and spouses inadvertently dislodge power plugs on a regular basis. It is a fact of life. Ensure that your power cord is firmly plugged into the wall (or surge protector) and into the back of the computer.
2. There could be a problem with your power circuit in the wall your computer is plugged into. Ensure this is not the case by switching on another electric device that is plugged into the same outlet or power strip as your computer. If this isn't an option, try plugging the computer into an opposite wall or in a different room.
3. The problem could be in the power cord itself. Swap it out for a different one - the cord style is very common.
4. If you've eliminated all of those possible causes, it is time to look into the computer itself. More than likely the problem is a failed power supply. The power supply is the big box (usually gray) located in the back of the computer. It is what the cord from the wall plugs into. If you are an intermediate or advanced user that feels comfortable working around the inside of a computer case, I recommend opening up the case and making sure the power supply is correctly connected to the internal components. If the problem isn't a loose connector, remove the power supply, check the wattage (sticker on the side), buy a comparable one and install it.
This (possible) solution will only cost you between $20 and $40 (for a normal desktop PC in the home or office), and if it ends up not solving your problem, at least you have a good power supply that you can use later on. If you would like to try this but aren't so familiar with the inside of a computer, I recommend this article on installing a power supply.