A router is a network device that connects one network (such as your internal home network) to another network (such as the internet). The wide area network (WAN) port of the router connects to the internet (DSL modem, Cable modem, etc.) while the other ports connect to the internal Ethernet network. Routers usually include a built-in switch so that there are multiple ports for internal network connection.
The router does the address translation and some firewall protection to isolate the internal network, while still allowing it access to the internet.