What is an ADSL modem
An ADSL modem is a device that transmits and receives data over the plain old telephone system's (POTS) analogue lines. It allows your PC to connect to your ISP using the Internet Protocol. The modem acts as a "bridge" between your PC and your ISP. The dynamic or static public IP address which your modem obtains from your ISP is passed on to your PC. Your ADSL modem forwards packets based on hardware level MAC addresses between your ISP's router (and the Internet) and your PC.
What is a router?
A router is a device which acts as an interface between two networks. It forwards packets based on network level addresses (Internet Protocol addresses in this case) between your ISP's router and either your LAN or your single PC. A router learns more about the networks to which it is connected and can be more selective about the packets it passes on. A router rejects packets unless they match predefined attributes (eg specific protocols or destination network addresses). A router can also select the best route for packets to take in large interconnected networks.