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OS X Mountain Lion needs Mac OS X Snow Leopard at least to upgrade from, so you have to upgrade first to Snow Leopard.
The disc will install Mac OS X 10.6.3, so open > Software Update to install 10.6.8. After updating to Mac OS X 10.6.8, you will see the App Store on the Dock, so first check if your Mac is supported > http://www.apple.com/osx/specs If it's supported, just open App Store and purchase OS X Mountain Lion. When the download finishes, the Mountain Lion installer will show up, so follow the steps to upgrade to Mountain Lion.
Check that your apps are supported > http://www.roaringapps.com