The network cable running into the back of your computer, along with the network cables running into all of the computers in your office, tie together, along with your servers and gateways into a wiring concentrator; usually a hub or a switch, sometimes a router.
A hub simply takes the signal coming off of one cable and rebroadcasts it down all the others. Sometimes two or more computers broadcast at the same time so a hub has simple logic which looks for these data "collisions" and coordinates the resends when they occur. Hubs are cheap, but seldom used today in high-load environments due to their vulnerability to collisions.
A switch, on the other hand, has additional logic that learns, on a higher network level, who it is directly connected to. When a data "frame" comes down the cable, it reads the intended destination, and sends it on only to that destination. By doing this, collissions are eliminated, as are their resulting wait-and-resends. This results in a much higer performance network, especially under heavy load when collissions can gridlock a traditional hub.
Routing equipment operates in a similar fashion to switching equipment but on the next higher networking level. Routing equipment determines the proper routing of data based on programmable, logical, routes instead of simply looking at a packet's destination and assuming it can be delivered locally. When you see depictions of the Internet or WWW, as a vast web, connecting many nodes together, the routing equipment is represented by the nodes. Routing equipment makes communication between networks possible.