It’s something else. Single-Player/Co-op is hosted locally, ie. on your computer. A private server is a separately running service where players can join at will (as long as the server is on), which means you don’t need to be present for other to play.

You can rent a server from an online service (such as G-Portal, which is where Funcom’s official servers reside), or you can install the private server software on your own computer.

But I don’t have any personal experience with that, so I can’t answer your question on what sort of hardware that requires.