You should need to press your Mac's power button only once in order to turn it on. After doing this, you should hear the boot chimes, then see it load OS X, and drop you to the log-in window. However, what do you do if this doesn't happen, and you neither see nor hear any activity from the system?
If this should happen to you, there are several things you can do before taking your system in for servicing.
First, check for a proper power source. If your system does not have a battery, test the wall outlet by plugging a lamp or other device into the socket into which your Mac was plugged, or try another socket altogether. Or just try using another outlet to see if that will help.
Sometimes a faulty wall socket could power a simple device like a lamp, but fail to run a more complex device like a computer's power supply. One way to test this is to use a socket tester (purchased for a few dollars at your local hardware store) to ensure the socket is properly wired.
If your Mac has a battery and won't start, try letting it stay plugged into a power outlet for a short while. Sometimes if batteries are completely dead, the computer will not start up. Charging the system for 5 to 10 minutes may help.
Finally, the chassis buttons and power of the system are managed by the computer's System Management Controller (SMC), which, if experiencing problems, may result in the system not responding to the pressing of the power button. If this is the case, then a quick reset of the SMC should fix the problem. The process for doing this is specific to different systems, but can be looked up in this Apple Support article.
If none of these options gets your computer to power up, then either the button itself, the power supply, or another component may be broken and will need to be repaired by a technician.