This is very much how a telephone switch places telephone calls. There is no rule about how many switches can be on a network, and similar to a bridge, a switch breaks up a network and allows 4 repeaters on each port. Routing software looks at protocol information in a packet of data to decide if that data should go to a network directly attached to that router, or a network further away. If it should go to one further away, the router figures out what other router is the next step in the path. It is also possible to route other network layer protocols such as the DDP protocol in AppleTalk or the IPX protocol in Novell's Netware.
Most companys selling routers sell both the hardware and the software all in one package.
The computer may or may not have specialized hardware. This may just be a PC-Clone 486 running a mail gateway and doing something else, or it could be a dedicated machine. A GatorStar GXR is a specific example, there are other products that do similar things. ATM using 2 pair. It seems to be the most versatile option currently on the market.