Dropbox accounts are completely free for up to 2GB of data, so start by dragging your most important, and most frequently used files to your Dropbox. If you outgrow the 2GB limit, you can add space for free by doing things like recommending the service to a friend or following Dropbox on Twitter. Or you can pay for more storage, starting at $10 per month for 100GB. Dropbox even offers group plans for businesses.
Thats it. The files upload (and continue syncing) in the background as you use your Mac, and you dont have to change anything about your workflow except the folder where you look for some of your files. You can freely open, save, and otherwise modify anything in your Dropbox, because as far as your Mac is concerned, its just another folder. And once your files are synced to Dropbox, you can access them using nearly any device that can connect to the Internetall you need is the Dropbox software or a Web browser.
I recommend Dropbox because it's worked amazingly well for me during the several years Ive used it. But you can choose from a great many other services with comparable features, including Google Drive, Microsoft SkyDrive, SpiderOak, SugarSync, Wuala, to name but a few.
Dropbox and similar cloud-syncing folders are not a good place to put iPhoto or iTunes libraries, because the libraries are likely to be quite large and may not operate correctly when synced between Macs. However, you can achieve a partial online backup of your most recent photos from iPhoto or Aperture by using iClouds Photo Stream feature (which you can enable in the iCloud pane of System Preferences); and if you subscribe to Apples $25-per-year iTunes Match service, youll get what amounts to online backup of your music (as long as your library of songs not purchased from the iTunes Store doesnt exceed 25,000).
Sync it to the cloud
Dropbox and similar services are great for stand-alone files, but for personal data such as email, contacts, and calendars you're better off with a service that keeps a master copy of your data in the cloud and syncs that copy to your Mac and iOS devices. If you still store contacts and calendars locally and rely on POP for email, consider moving your data to the cloud. Its much easier than it sounds!
For email, any IMAP server will do the trick, including iCloud and Gmail. The IMAP server always has a copy of your stored email, so if something happens to your Mac, you can download a fresh copy easily. Although IMAP doesnt qualify as a true backup, since you cant recover messages that have been deleted permanently from the server, its much better than nothing. If youre still using POP, read How to convert a POP email account to IMAP for guidance on switching to IMAP.