Router:
A router is a device that forwards data packets along networks. A router is connected to at least two networks, commonly two LANs or WANs or a LAN and its ISP's network. Routers are located at gateways, the places where two or more networks connect.
Using assigned numbers, the router keeps track of which computer made a request to the Internet in order to send responses back to that computer only.
Modem:
A modem is a hardware device that allows a computer to send and receive data over a telephone line or a cable or satellite connection.
Modems are generally classified by the maximum amount of data they can send in a given unit of time, usually expressed in "bps"
This is a small device that connects to your internet service provider (ISP) to tap into all that internet goodness. The connection is made via a cable (for cable or fiber internet) or phone line (DSL) from outside your house that plugs into the back of your modem. Your modem shares this connection with a computer or a router via an Ethernet cable.
Do I need a modem?
Without a modem your router will only allow you to connect to a LAN (local area network). The modem is your connection to the Internet.
Well, a modem just connects to your isp so that you can connect to the internet. Something that is just a modem will likely only have a port for the phoneline and an ethernet port.
Do I need a router?
Picking a modem is only half the battle, because they typically provide connections for just a single, wired device. If you want to go wireless, you're going to need a router, which is a networking device that lets you share your modem's internet connection with all your devices. Routers connect to your modem via an Ethernet cable and pass that internet connection on to other devices in your house either via an additional Ethernet cable or wirelessly over a Wi-Fi network.
If you have multiple devices to connect or want wifi you need a router to plug into the modem. A lot of the time the modem your isp gives you will have a cheap wifi router in it already.
Conclusion:
A router is used to connect devices together to make a network. Called a LAN or Local Area Network.
A modem is used to connect devices to the Internet. Called a WAN or Wide Area Network.