There are hundreds of backup solutions out there, including hardware like the Time Capsule, external hard drives, or a NAS. There are also tons of services for offsite backup, like Carbonite or Sky Drive. Here are a few of the services I used today as I did my backup.
Dropbox allows you to store your documents in the cloud, giving you 2GB for free with an additional 1GB for jumping through a few hoops. There are paid plans that allow more storage as well. One of the best things about it, though, is the fact that you can install the Dropbox software onto multiple computers, allowing you to keep documents and files synced across your different machines, including your iOS or Android device.
Amazon's recently announced Cloud Drive offers 5GB of free storage with an additional 15GB if you download an album from the Amazon MP3 store (mine cost a total of $2.89 for 20GB of storage!). While it can be use to store any type of file (2GB file size limit), it shines with music. The ability to store your music and stream to any computer or mobile device is extremely compelling.
I'm probably a special use case, here. All of my email, documents, etc. live on Google's servers (I haven't had an office suite installed on my computer in years). The Data Liberation Front offers tools and instructions to help you export and import your data to and from Google's services. From Gmail to Docs to Calendar, their goal is to make sure you retain control of your data.
Backupify is a service that will easily backup your online data. It offers weekly backups with 2GB of storage for free, with plans to increase storage, frequency, and services. Backupify is a great way to backup your Twitter, Facebook, LinkedIn, Blogger, or a plethora of others
No matter what service you use, the most important thing to remember is that you need to have multiple backups. My suggestion is to have your files stored on at least two computers (Dropbox is really helpful for that), an external drive that is not in close proximity to either of those computers, and a cloud-based service. That way, unless there's an apocalyptic event, you should be able to get your files back if you lose one copy of them.
What's your favorite hardware, software, or cloud-based service for backing up? Let me know in the comments.