Modem and router are two different networking hardware equipments. Modem is a modulation device, that is, a modem can modulate and demodulate data. We know that data can be either analog or digital. In computer networking, the data to be used in the computer needs to be digital but the data that is used in the networking can be analog.
For example, while connecting to the Internet by using a telephone line, the data in the networking process, that is, the data from the telephone line are analog. Now, the analog data needs to be converted to digital for processing in computer and digital data needs to be converted to analog to send from computer via the telephone line again. This conversation is the modulation and demodulation processes and the job of modem is basically to convert analog data to digital and digital data to analog.
The job of router, however, is quite different. A router does not change one kind of data to another kind. It simply forwards the data. Router works on a group of data, which are normally called as packets, and forwards the packets across the network.
By: Eric