Definition:
Routers are specialized computers that send messages from Internet users to their destinations among millions of pathways. Without the router, using the Internet would not be possible. In simple terms, routers are the traffic directors for our information highway.
Details:
The router's job is to pass what ever information specified between networks. Routers also protect networks in a way, by preventing data from entering and possibly flooding the wrong network. The data is transmitted in the form of small information packets.
Depending on what what job is to be performed, the type of router may vary.

For example, for your own personal network, you may use a small router to communicate between computers like the one to the left.
There are slightly larger routers that may handle more connections and serve as a security protocol as well. These types of routers are usually used within businesses, and are usually stand-alone, meaning they don't need to run on software.
And last, but not least, the largest routers handle major traffic points on the world wide web and have more in common with a super computer

than your average wireless router. This type of router is depicted to the right.
Routers may also use
MAC addresses to help determine where to send the information packets.
The wireless router depicted above may also be connected to a
modem to provide wireless Internet to capable devices.
In all, routers form the backbone of the Internet by connecting all of the networks together and directing the information being sent and received.
Related Links:
How Stuff Works: How Routers Work