Upgrade OS X Lion (10.7.4) to OS X Mountain Lion
Actually this is pretty simple.

Though you can install Mountain Lion right over (upgrade) Lion without doing any backups at all a more prudent method would be to backup first.

To back up your computer you will need an external hard drive (USB, Firewire, or Thunderbolt. USB would likely be cheaper but a bit slower) that is a little bigger than the HD that you are upgrading.

If you plug the USB drive into your Mac, Time Machine will automatically ask you if you want to use it as a backup drive (not sure about the actual wording...) You can allow Time Machine to proceed and back up your whole computer. Once it is done, un plug it and proceed with the upgrade.

Alternatively you could use a commercial backup program to do the same thing. I have used both Carbon Copy Cloner and SuperDuper to make backups of my drives. Bot work really well and are not expensive. The advantage with these is that they make a BOOTABLE backup image of your hard drive, where Time Machine will need you to install the O/S to then do a restore of your data.

Once you have made your backup (and tested it by checking a number of files or booting from the backup and verifying it all works) just go to the App Store (in your Apple Menu) and purchase Mountain Lion. It's a BIG download (4+ GB) and will take a while, when done it will automatically prompt you to start the installation. Just follow the prompts. Your Mac will reboot and proceed with the upgrade. It is likely that you will not need your backup as these upgrades usually go quite well.

There are some caveats. If you have modified your O/S or used applications that make changes to the O/S you might want to remove them before upgrading. You may also want to check that the versions of any applications you installed are compatible with Mountain Lion. That info will be on the manufacturers website. Basically you need to be aware that it is possible that something that worked fine under Lion will NOT work under Mountain Lion. Only you can make that determination.

Some people take the pre-install procedure a bit farther than just a backup. It is not uncommon to boot your Mac from the original install media and run disk utility and run a "repair disk" and "repair permissions" on it before installing. You could also use a maintenance tool like Onyx to do some basic repairs and maintenance on your system before upgrading as well. Some people won't upgrade before doing either or both of this things, some people just download and install the new O/S with no special prep or backup.

Really it's your choice.