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A hub is a device that connects multiple computers together. The speed is limited to 10 Mbits, and only 1 PC may transmit or receive at a time regadless of the number of ports.
A switch is a device that connects multiple computers to each other. The speed is 100Mbits, and every computer can transmit and receive at full bandwidth
A router is a device that sits between an outside connection and an internal network. To the outside world the router is the PC, it then "routes" the signal to the proper PC (via internal IP address) on the network. Other than the outside connection (and usually a firewall) it works like a switch. This is often used for sharing internet connections across multiple PC's.

I believe Hubs go up to 100Mbits and how it works is that bandwidth has to be shared among the computers connected to it. So 1 100MBits connection split to 10 users = 10Mbits each user, not too bad but one 100Mbits connection split between 100 computers = 1MBits per computer.

Let's try this one again.
A hub is a single broadcast domain which can be 10 or 100mbs. This means what is sent to one port is sent to all ports.
A switch has multiple broadcast domains and can be 10/100/1000mbs. It is intellegent in that it looks at the mac address contained in the header of the packet. It is able to build and retain a mac address table so that it can send the packet to exactly the correct port and not all the ports. This is what makes it faster then a hub. You can also do a number of other things like VPNs depending on the switch.
Since both are ethernet and use Csma/cd [carrier sense multiple access/collision detectiion] the mbs speed really isn't "shared" or divided up evenly. It is timesliced. So one wkst can only talk at a time which means at 10mb you have x timeslices but at 100mb you have 10x timeslices. Its the difference between a 4 lane freeway at 55mph and a 8 lane freeway at 55mph. Same speed but more cars, more timeslices, can get on the freeway. Don't get fooled by the 10mb vs 100mb as faster. It doesn't work that way. 100mb has more bandwidth [lanes].
A router is an intellegent device that can decide what to do with a packet. Should it drop it or should it "route" it. For example lets say you have a router with a ethernet port connected to your internal lan, a T1 port connected to the internet and a Frame Relay connecting to another remote site. Depending on how you configured it the router can send the lan based email you sent for your lan based wkst to the FR port and the IE web page request from the same wkst to the internet via the T1. A router can route any protocol except netbios which is not a routable protocol. It doesn't contain enough header space to say where the packet is from or where it is going. Even this can be gotten around by encapsulating netbios in something else like ipx or tcp/ip.
If you want more info then this a google search or textbooks can give you volumes.

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