A router basically just "routes" your DSL or cable connection to multiple components or users. For example, a router can allow two PC users to effectively use one cable or DSL connection. It can also allow sharing of that same connection with as many devices as you wish depending on how many ports your router has. A 4-port router can allow for any 4 devices to share a single connection. You can have 2 PC\'s and an XBox and PS2 hooked up to a single network like I will soon have or for your single computer and all three consoles connected to it. You can even hook a laptop, 3 PCs, xbox, ps2, and a GCN to a single Internet connection if you wish with an 8-port router. Routers are just incredibly easy to setup and cost nothing for the investment it guarantees you.
You can go wireless if you wish, but those cost more and may perform slower than a standard network cable connection instead depending on any types of interferance, but it\'s all up to what you want. Network cables aren\'t that expensive in the first place and they are incredibly durable, much moreso than your standard phone line. If that still doesn\'t make sense, here\'s a nice little diagram of a similar setup I have, just with DSL and another PC instead of the Gamecube Internet connection:

Pretty simple. Even the software is a joke to use and takes no more than 2 minutes to setup on the main PC that controls the router itself and the rest of the items after you hook up your PC is just plug and play after that.