Any time you connect two networks, you must use a tool that translates the address and control data used by each network into values that the other network can understand. When the device simply examines the address on each packet and decides which packets to forward to the other network, it's a bridge. When the device examines the address (or routing) information in each packet and sends the packet along to its ultimate destination (which might be in yet another LAN far from the original), it's a router.
Note
Bridges and routers operate between two different networks. Don't confuse them with switches and hubs that distribute data within a network. However, many routers combine functions with a switch in a single device.
When you connect your LAN to the Internet, you add a router as a node on the LAN. This kind of router is sometimes called a gateway router because packets from your LAN must pass through it on their way to the Internet (and, of course, packets from the Internet also pass through the router on their way to your LAN). The most common gateway routers are specifically designed to supply the right kind of signaling and address conversion required by DSL or cable TV connections.