You have no idea how frustrating it is to try to explain the difference between dial-up networking and REAL networking, when the be-damned marketers insist on calling a router by the misbegotten name "cable modem."
Okay, there is a "modulator/demodulator" in there. It is converting digital to analog to put it on the cable (and at the other end there is a strange device called a DSLAM to convert it back again).
But from the point of view of your LAN, it's a router. Router, router, router! It connects two different networks together. It has two IP addresses: an inside and an outside. You should not think of it as a modem -- at least, not primarily.