There are many ways of defining what a Router is. Here are two:

Definition 1:

A device or setup that finds the best route between any two networks, even if there are several networks to traverse. Like bridges, remote sites can be connected using routers over dedicated or switched lines to create WANs. - www.ohsu.edu/vcs/glossary/

Definition 2:

A device that determines the next network point to which a data packet should be forwarded enroute toward its destination. The router is connected to at least two networks and determines which way to send each data packet based on its current understanding of the state of the networks it is connected to. Routers create or maintain a table of the available routes and use this information to determine the best route for a given data packet. - www.ml-ip.com/html/support/glossary.html

Simply put, a Router is a device that routes the signal from one network to another or from one computer to another on any given network, allowing transfer of information. Routers do not have definite numbers of ports as it all depends on where exactly it is going to be used. Some Routers have as little as three (3) ports (one input and two outputs) to more than ten. Some have multiple input ports also. Prices also vary depending on the features and brand name. Some of the top-of-the-line names include NetGear and Cisco.