There are a few ways for you to turn your computer on remotely. If you want to do this on a networked computer on the same network you can use Wake On LAN assuming your computer's network card supports it.
If you want to do this from a remote location then it can be done over the internet a few ways. You can use the Wake On LAN method or Wake On Ring method which is similar except it uses a modem instead of your network card. You most likely will have to have a static IP address for the location where the computer you want to turn on is located as well as a method of translating the public static IP address to the local internal address. Check out this link for more details.
There are also other hardware devices that will do the same thing as WOL but with more advanced features.