GREAT resource at the Xbox.com web site: All about routers in home networking:

What the heck is a router, what does it do, and why should I care?

A router is a device that connects two computer networks together so that they can talk to each other. A router allows your Xbox and one or more PCs—your “home network”—to connect to the Internet at the same time.

…Different manufacturers have different names for these devices, including home gateway, cable/DSL router, broadband router, broadband base station, and more. These are generally different names for the same thing—all provide the same basic features you need to share a connection.