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The modem is what connects us to the internet. A Modem is a combined device for modulating and demodulating. To modulate, you exert a controlling influence; meaning that the Modem modulates the analog carrier signal so it can encode digital information. Demodulating is to extract from its carrier. That means that the transmitted information being produced is getting decoded by the Modem. (Modulate/Demodulate = Mod + Dem = Modem)

Modem connections
The Modem goal is to change the form of its electrical signals so that the information can be sent from computer to computer through a telephone line. 
Cable-modem-intro

This is a modern Modem.

The types of information that it could send over the line would be e-mail, internet access, and fax transmissions. The average speed of transmission is around 56 kilobits per second. Over time, it gets updated to allow more and more be sent over the line, at a rate twice as fast as it use too. ISDN lines are what raise the rate up to twice the speed, but things like cable modems and DSL lines can transmit over a million bits per second.
Router technology hw

This is a router.

People often get the Modem and the Router confused. Both the Modem and Router are devices used to connect your computers to the internet, but the difference is that the Router directs information from Modems to things like computers, iPods, e-readers, and other internet accessible devices. The Modem encodes and decodes the data, so that the router can direct it to the devices that utilize the information.

The history behind the modem goes all the way back to the 1950's. The modem was originally created to transmit information for the Air Defenses. They would communicate over the Public Switched Telephone Network (PSTN). PSTN is the world's public circuit-switched telephoned networks. The Modem was finally first used publically when it was commercially manufactured by AT&T. It was called "The Bell 103", and it was, may I say, quite slow. 

http://inventors.about.com/library/inventors/blmodem.htm

http://www.merriam-webster.com/dictionary/modem

http://www.diffen.com/difference/Modem_vs_Router

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