Three routers, A, B, C. Each router has a T3 line which is run by the telephone company, passes through the phone company's network (which is just a bunch of T3's and other larger lines that don't care that this is IP traffic, they're just passing the physical signal), connects to the T3 connected to another router.
Each router has a computer connected to it. Computer A1 sends to computer B1. Viola. The Internet bandwidth.
"Bandwidth" refers to how much data a particular type of connection is capable of passing. The type of line is often dependent on the equipment plugged into it. For instance an OC3 and an OC192 both use a fibre optic cable for their traffic, but an OC192's interface card is able to signal faster and muliplexes across more frequencies so that it can pass more data per second.
The bandwidth is "created" by connecting two devices via a certain type of cable, using a certain type of interface card. The bandwidth between two points is limited by the equipment between them. So a server may have a 100Mbps Ethernet connection to its router, and your computer may have the same, but if the two routers connect using a dialup line, you're only going to have a dialup line's worth of bandwidth between you and the server.
A network is simply two or more computers, connected via any of a myriad of types of communication systems, that pass data to each other. This could be dialup connections, satellite, DSL, cable modem, T1, T3, OC192, telegraph, ISDN, ham radio. They could be using TCP/IP, IPX, anything, as long as both ends are programmed to understand what the signals mean.
Once you get to a size where "routing" becomes necessary, then the devices in between have to be able to understand the signals as well. That's why a dumb hub can work with token ring, Fast Ethernet, anything that connects with an RJ45 connector -- it just replicates the electrical signal. You could just connect a couple of home-built telegraph machines to a hub and communicate with them. But a switch requires that a MAC address be assigned and included in all packets of data, because it has to know where each machine is located. A router goes even further, so an IP address has to be assigned to each machine as well and included in the data packets. (Other protocols can be used as well of course, IP is just the most common now. Not all routers support all protocols.)