Shoutbox

A router is a router and will work in (almost) ant network. Commonly, routers are integrated with modens in the same box, and the modem connected directly to one end of the router, and the rest of the network goes in the other side of the router.

The router fulctionality can be utilised (in most cases) without using it as a modem as well.

You cannot however use your ADSL modem (combined with router or otherwise) for a cable internet connection. Cable is a dedicated digital line which carries heaps of TV and internet signals. ADSL uses a reserved frequency of your phone line between you and your exchange, and then into a DLSAM. The technologies are completely different. The plug wouldn't even fit if you used a hammer.

If you want to use only the router functionality, and you have a cable modem with its own login client, and an ethernet connection, that, and the other computers could be connected through the router, I think, it would need NAT and I don't know whether the (cable) modem would actually work like this.

They might use PPPoE, but the actual signal carrying PPPoE standard traffic is sent as cable signals or as ADSL signals, and they are both different, there are even different types of ADSL connection.