A router is normally used to connect two networks, determining (by means of IP addresses) which packets of data are intended for the local network (eg if you are copying a file from one PC to another in the same home/ office) and which are meant for the 'remote' network (also called the WAN or Wide Area Network).
For a big business, the WAN might be a network linking offices in different cities. For most of us, the 'WAN' is a link to our ISP and thence to/from the 'internet'.
The router does some important work in tracking which local PCs have made requests to remote network addresses, and then when data comes back, it correctly sends it to the correct local PC.
That might sound easy (compare person A making a call to extension A at some remote service, and person B calls extension B... everything is plain, when an answer comes back, if it came from extension B, it goes to person B locally) but if two or more people in the house use a single website (eg BBC iPlayer or Facebook) then the router has to inspect the data to see which machine needs each packet sent up the line from the ISP...
Sometimes when more than one person uses a Torrent, the number of connections that the router has to track causes it to run out of memory and it may either crash, or act dead and need to be rebooted.
HTH. Perhaps Wikipedia has a shorter (or more technical!) description for anyone who wants another explanation...