You're at the coffee shop. You need to access a file that has information about your work, but as you attempt to find the file you realize that you've made a mistake. You saved the work on your desktop computer, but you only have your laptop with you. You have no way to access the file.
This problem, and others like it, are increasingly common today. It isn't unusual to own both a desktop and a laptop computer, and a smartphone on top of that.
The convenience of being able to relax at a coffee shop while working on your laptop and checking movie times on your smartphone is attractive, but it makes organizing your files difficult. You may find yourself frequently emailing files to yourself, or dragging files back and forth from a USB drive. This only works as well as your own memory: if you forget to transfer files you will find yourself up a creek without a paddle.
Dropbox solves these issues by making it easy to synchronize files between computers. Installing Dropbox creates a special folder on your computer. Anything that you put in this folder is automatically synchronized with any other computer on which you've installed the service. The files you drop in for synchronization are also located on a remote server, which means you can download news files even when all of your other devices are turned off or offline.