As FTP is tagged, which is quite an acceptable approach to the file transfer, all you need is to have FTP server on one of the hosts, and you can develop a client part of FTP transfer using the class
(Would you need to develop the server-side of FTP as well? I don't think you will need it, but you can find an appropriate pure .NET library on CodeProject and elsewhere. FTP is the application-level protocol over TCP, which is fully and natively implemented in .NET FCL, so this is not a problem, too.)
Probably you did not find how to copy a file, because you did not learn all the request types:
To copy a file, you use either the method
Also, you should not forget that the FTP server can configure different access privileges for different users, so you have to take care about proper configuration.
You can find the upload code sample here: http://msdn.microsoft.com/en-us/library/ms229715.aspx