It’s a way of syncing files between computers using the “cloud”.
When I was putting together the details for my Sicily trip last summer I had a document that had everything – every confirmation number, contact info for the vacation rentals, car rental info, everything that I might ever need. I used drop box to move/sync it between my work, home, tablet, husbands computer.
Basically, you install the app on all the devices you want to sync between. Then if you update one of the files in your dropbox folder, it will get synced to the other machines when you log onto them.