I am trying to write a program that sends basic text files from one computer to another using C++ over the internet (or a LAN). I was looking into Winsock but everything that I read made it seem like it was only for communication between a server and a client. I am not sure if this is what I am trying to do or if I am looking at a different problem.
EDIT: Thanks for the great answers guys. I kind of feel bad for having to choose a best one out of the lot.
In short: Yes, sockets are the way to go.
"Server" and "client" in the sockets sense are generic terms - your system is running a variety of both at any given time. For example, your web browser operates as an HTTP client (where HTTP is a text-bounded mostly-synchronous protocol running over TCP/IP); Windows file sharing and remote desktop are "servers" that other systems connect into.
The distinction between server and client in the sockets sense (really in the TCP/IP sense) is that a "server" socket listens for incoming connections from remote systems (refer to MSDN or man pages on bind(), listen(), accept()) whereas a "client" socket creates outgoing connections to remote systems (connect()).
Beware that sockets programming can be a bit tricky. What you describe is a good learning exercise, but if you're trying to get something done quickly, you might look at existing tools such as netcat.