It depends on the router. If it's a combined modem/router, it connects straight to your normal telephone/cable connection that you use now.
If it's just a router, it will connect to your existing modem.
Once it is set-up, the router provides the "gateway" to the Internet. Each computer then connects wirelessly to the router.
If your existing desktop machine does not have a wireless capability, you will need to buy a suitable wireless card to enable it to connect to the router.