There's a hub, which allow several computers in one physical location to 'talk' to each other. Great for sharing files or fragfests. Then there's switched hubs, which do the same, but instead of parceling out the network speed (dependent upon the speed of your NIC or adapter), each line gets full speed.
Now on to routers. Like the hub, there is an option to network several computers in-house. You can also connect your external modem to share the internet connection as well. Note: this parcels out the overall speed to each computer connected to the router. A router aslo does the nefarious task of providing a single visible IP, while hiding the computers connected to it. Most ISP don't like this, since those that bill per user only see one user, not 2 or more. You can also get a switched router, which is just like the switched hub for internal NIC speeds, but there is still a sharing of the internet speed.
Currently, we only have an internal dialup modem, but I'm on a laptop while my ladyfriend is on the desktop. We use ICS (internet connection sharing) through USBNet (USB Ethernet adapter). This works pretty good for file sharing (a boon when we need to reformat a drive, since all archived setup files are on a partitioned drive), online games, internet sharing, and intranet games (fragfests). Best $20 I ever spent.