A modem translate the signal from the outer world (dialup, cable, ADSL, satellite) into something your computer understands and can connect to (using a RJ45, USB or serial plug).

A router can be used to connect multiple computers (LAN) to the modem (WAN), so they can use Internet. And, implicitely, connect the multiple PC's to each other so they can use resources (files, printers) from each other.

To make it more complex: a modem with built-in router can be called a "multi-PC modem".

Kees