One of the easiest ways to back up the files on your computer and keep them safe is to put them on a flash drive. What's a flash drive? Geez Louise, where have you been the last 10 years?
Flash drives, also known as thumb drives, are like itsy-bitsy hard drives. They're small enough to carry around on a keychain, or around your neck (some even come with lanyards so you can do just that). You just plug them into the USB port on your computer and they act like a hard drive.
When they were first available they wouldn't hold much data, but I saw one on Amazon just now that will hold 128GB. That's bigger than the hard drive on my last Dell desktop. They're asking $375 for it, but the smaller ones are dirt cheap. You can get a 2GB model for under $10. That's a lot of storage.
They have a lot of advantages over CDs and DVDs for backup, too. For one thing, they're solid-state, which means they don't have any moving parts. So they can get knocked around quite a bit without breaking. I've lost CDs by accidentally scratching them, and one by leaving a post-it note on top and having the surface of the disk peel away when I took it off. Who knew? Flash drives are also generally formatted so that you can use the same one on both Macs and PCs, which is handy.
You can buy them at almost any place that sells office supplies. My local hardware store sells them. They're all about the same, so don't worry about which brand to get. Get the cheapest one they have. And back up your stuff.