Routers are devices used to connect networks together. Large organizations will have larger devices in their rooms where the telephone line or Internet net connection comes in that are Routers. They will only have a few connections on them but you can program them to control who is allowed in your network and what types of requests can come in your network. you can also program them to say what is not allowed in your network or what is not allowed to leave.
If you have a connection to the Internet using DSL or cable then you have a router although it may be called differently. these boxes are small and will only have a few connections also.
Usually you will see them called DSL modems or cable modems. They really are not modems as modems really convert digital signals to analog signals (your computer uses digital signals and telephone used/uses analog. With dialup you have a modem. People got used to the box between the telephone line and the computer being a modem so they refer to the box between the cable or DSL line and the computer as a modem. In the case of DSL and cable you have a digital signal on the cable or DSL (the D in DSL is digital) and the computer is still digital so it is not a modem really.
The DSL/cable modem is really a modem as it is connecting your network (your computers at your house, one or more, is a network and you are connecting to your ISP (cable or DSL) network. A router connects the networks together. You can do limited rule setting in these personal routers.
Special note: This is the 100th question answer article I have done and I continue to amaze myself weekly on finding another subject to tell my readers about. I appreciate the newspapers who carry this article, and especially the Swainsboro Forest Blade for giving me the original opportunity. To the Blade, the Messenger and the Catoosa News and all my readers thank you for the encouragement over the last two years. There is still plenty of material for the next 100.