A .dmg file is a copy of a physical disk, such as a CD or DVD, contained in a single file. The file extension means "Disk Image" and is proprietary to Apple. The operating system treats this file as it would a physical disk. Normally OS X will automatically mount a .dmg file when you double click it, however should this fail you can manually mount an image using tools that come with the operating system.
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Open "Disk Utility" from your Applications list. It should be in a folder called Utilities.
Click "Mount" at the top and select the .dmg file you want to open. Once mounted it will appear on your desktop as an icon.
Double-click the icon of the image to open it.