I've been asked by several of my non-technical friends how I back up my computer & files. Here's a quick rundown.
Backblaze (referral link) - I would probably consider this my most important backup software. There is a desktop client for Mac and Windows, as well native mobile clients for iOS and Android. The price is $5/month or $50/year for unlimited data. This includes external hard drives. Yes, you read that right. You can download any of your files from the last 30 days from a browser, or should catastrophe strike, a flash or external drive will be send to you for an additional fee.
Dropbox (referral link) - this is a common backup system many people already know about. I place most of my documents there as well as my photos. I have had an account for years, but now the only reason I really keep it around is for photos. RAW takes up quite a bit of space. You get a free month if you use my referral link. Alternatives: box.net
Time Machine - this is built-in for OS X users (it does require an external drive). I recommend it for disaster recovery more than anything else. If your machine dies, get a new one and hook it to your external hard drive and restore.
Nerd-type backup systems
Github - or some damn offsite source repository. This shouldn't need to be explained.
rsync - a standard linux utility to sync directories between two machines. Once I get a server, I'll probably rsync files from it (or vice-versa) to my laptop to get under the Backblaze backup.