Mac OS X: Backing up my Mac
The device in the photo is called a USB SuperDrive. It is a CD/DVD reader made by Apple. I wouldn't recommend making a backup on CD or DVD discs. But you'll use the SuperDrive to run the update on the update disc you have. Do not overwrite that update disc or make any changes to it, or it will become unusable.

I recommend using a USB external hard drive to back up your aunt's files. You can buy an external hard drive at Apple, Amazon or any computer store. 1TB is the standard amount of storage these days, and it is unlikely that your aunt has more than that to be backed up.

Finally, I highly recommend using Time Machine (an application that is already installed on your aunt's Mac) for backing up. It backs up the whole system (photos, music, applications, documents, everything), and it is the safest proprietary backup method. I recommend doing this before attempting an update. 

You can use an external hard drive with Time Machine to backup the system. When you plug in an external hard drive for the first time, the Mac usually asks if you want to use it with Time Machine.

If your aunt really doesn't care about anything other than a few files, you can back them up to a free service like Dropbox designed to hold small amounts of data. Just be aware that if something goes wrong during the update, you risk losing everything that isn't backed up.

Short of this, Apple has really excellent technical support, and while I can't make promises for them I've found that they're more than willing to explain these concepts in more depth. (They just can't perform a backup for you.)