A Router is a network switch which allows you to connect one or more devices to a single ADSL (internet) connection. One is normally bundled in free when you sign up to any Internet Service Provider (ISP) and can be either wired or wireless - or normally, these days, both. ISPs often refer to a router as an Internet Hub.
The standard routers provided by ISPs today normally comprise one ADSL input port and 4 ethernet connection ports. The photo below shows a view of the rear of a standard ADSL router:
Note the following:
The ADSL port is the one outlined grey one on the right hand side; this is where the output from the phone/ADSL socket is plugged
The four ethernet ports are outlined in yellow, in a single block in the middle of the rear panel. This is where your PCs or DLNA devices are plugged, via a standard ethernet cable with an RJ45 connector at each end
The power connector is shown on the extreme left of the rear panel, with a recessed "Reset" button just to the right of it
The router's job is to pass packets between any devices attached to it locally and the wider network. Some common home network configurations are given in our Server Guide.
Note: a router can also be used as a Network Switch. However, routers are generally more sophisticated (-and expensive) and so it is really only worthwhile if you have an old router sitting around (-for example, one left over from an old ISP)