Do you have to hook the router up to a computer?

No - a router can connect to a computer, another router, a switch, a hub, a bridge or a modem. If you are connecting to the internet via DSL or cable company you will need a modem/router to connect to them. You then attach your computer (or computers) to that router.

Routers are useful because they make decisions about which way a packet of data goes - this network or this network? If you have one network that doesn't go anywhere (connect to the Internet or another network) a router is of no use to you and a switch generally does wonderfully and is cheaper than a router.