Dropbox is a service that puts a folder on your computer called "My Dropbox". Anything you place in that folder is instantly backed up online and also synced to any other computer where you have Dropbox installed. You can access your files from any computer via the online interface at Dropbox.com, including every version of that file since you added it to your Dropbox folder. This means that if you ever accidentally overwrite your final term paper with a letter to your friend back home, you can just jump over to the Dropbox website and find a list of every revision you've made on that file. Simply choose the one you need and download it.
File backups are by no means the only useful feature of Dropbox. As I mentioned earlier, Dropbox syncs your files across any computer where you have installed it. So if you want files from your home computer to be accessible at work, just throw them in your Dropbox folder. It's that easy. Any changes you make at work will instantly be synchronized at home, and vice versa. Dropbox also offers mobile apps for Android [Market link], iOS [iTunes link] and (soon) Blackberry, so you can access your files on the go from your smartphone.
Dropbox also lets you share folders with any other Dropbox users. You just click the folder, click "Share" and enter the email address of the person you want to collaborate with. As soon as they accept your invitation, your folder will appear in their Dropbox folder and you'll be ready to go. Using this feature, I can work on my programming project from my laptop or desktop, my partner can use any of her computers, and we are still assured that we will always instantly have the latest version of our files.
Every user gets 2GB of space to start out with for their Dropbox, but you can expand this number up to 10GB by inviting friends to the service. Full disclosure: If you sign up for Dropbox using my referral link, I'll get an extra 250MB of storage. But on the bright side, so do you!
In case this is easier for you, here's the QR code for the Android version of the Dropbox app: