As of December 2015, OS X Mountain Lion 10.8 is no longer the newest version of Mac OS. As such, Apple does not provide a way for the users to upgrade from a previous version to Mountain Lion 10.8. However, users can either upgrade directly to the newest version, El Capitan, or perform a clean install to Mountain Lion 10.8, as of 2015.
To upgrade to El Capitan, visit the OS X El Capitan page on the Mac App Store. Click Download, and follow the instructions. Users without a broadband connection can visit a nearby Apple Store to download the update.
Users need to buy a copy of OS X Mountain Lion 10.8 to perform a clean install. It is available for purchase on Apple's online store and is delivered as a redemption code. The user must then use the redemption code in the Mac App Store to download the operating system. Apple notes that at least 2 gigabytes of memory and 8 gigabytes of available space is needed.
Once the download for Mountain Lion 10.8 is complete, find the downloaded installation file. Its name is Install OS X Mountain Lion.app, and it is likely to be in the Applications folder. Right click on it, and click Show Package Contents. Open Contents and then Shared Support. Shared Support contains a file called InstallESD.dmg. Launch Disk Utility, drag InstallESD.dmg into it, and write its contents to a USB disk. The Mac may ask for your user name and password when using Disk Utility.
To install Mountain Lion 10.8 with the USB recovery disk, reboot the Mac with the USB disk connected and hold the Command, Option and R keys after the startup sound plays. Release the keys when the Apple logo appears. When the recovery options are listed, select the USB recovery disk.