Your cable internet company will require a router so you can connect to their network(the router i mentioned is also called the cable modem).
After this box is installed, then no you don't have to use another router, you could use a proxy solution--like Win98se(ICS), or Win2000(ICS)...
A router is basically a hardware proxy with a firewall normally built in, or you could say a proxy is a software router, that sometimes includes a software firewall. (this is as simple as i can explain this)
So yes you require the cable modem(router) to connect, but you can use proxy software to share this connection.
Cable modems originally required a phone line to be used as the upstream connection--thats where the modem part came from.
(please correct any errors you find)