If you really just want to "send a file from my computer to someone else's computer" using C# then you may not be looking for true p2p. You can just use raw TCP. For this you need the remote computer to listen for a connection, your computer to open a connection to the remote computer, and start sending data.
There's a very basic example on how to do something like that here.
If you are actually looking for true P2P then you're best off using an existing P2P network (otherwise there will be nobody but you and your other computer on it). There are a few C# BitTorrent libraries around - for example BitSharp, TorrentNet. There is a whole question about BitTorrent libraries written in pure C#.
