Router and Modem are two different technologies, but often come in one equipment.
A MODEM is MOdulator DEModulator. Your Internet or network on TCP/IP protocols with Ethernet cabling & standards. If your ISP is using a different connectivity type e.g. Coaxial Cable (Internet and Cable TV together) or its on a telephone line (Internet + Phone) you will have to use a MODEM for converting your PC to Internet traffic digital signalling to ADSL etc... and back.
Routers function is to connect different networks (Layer 3 of OSI Network Model) together. When you take a internet connection from ISP, you get only single IP. So to use multiple devices like phone etc, you can put a Wireless Router. Your devices will connect to the router and you can enjoy internet on all devices. If you have a modem based internet, then router uplink can be connected to this modem.
