Dropbox is a utility that you install on your PCs or on your mobile phones that keeps your files synchronized across those devices. So say you have a work PC with all your work files on it and a home laptop. What if one day you want to work from home? Well, instead of transferring all your files over with a USB stick you can install Dropbox on both machines and it will automatically synchronize the files.
This is known as "cloud storage" and is a growing trend as people start to use more and more computing devices in their lives.