My guess is that's it's just a cloud system The originator of dropbox has a server computer that stores the data and a program (dropbox) on each computer that can upload/download data to/from the server.
How to write such a program in c/c++? The server side is probably running PHP. The client programs could be written using sockets. So if you don't know how to use sockets then that's the place to start.
boost::filesystem may be useful to you because its platform independent which means you write your program once for both MS-Windows and *nix operating systems. You'll have to download and install boost onto your local computer.