Modems Vs Routers
The Basics

Modems

A Modem will convert data for a router so it can send the data to another network. A broadband modem allows data to be sent down a phone line.
An IC-Air radio is also a modem as it allows data to be sent over the air. A 3G or 4G modem allows data to be sent over a cellular connection.

A typical  BT modem will look like this

Routers

A router connects at least two networks together by talking to other routers. A router normally instructs the modem what to do and when to do it. 
Some routers are modem routers as they have modems built in.
Some routers are firewall routers as they have a basic firewall built in.

 A typical router will look like this