First, are we talking wired or wireless? If it's wired, the simplest lan is at least two computers connected to a hub with no internet connection. More commonly, you'll see them connected to a router, and the router's wan port connected to a cable or DSL modem. This way is much more secure, because the LAN passes all data to all the computers, assuming that only the intended recipient will look at it, but this leaves it open to sniffing. I'm not going to go into that, but suffice it to say that it's a risk if you have untrustworthy people on your network with a hub.
Wireless, you just connect the cable or DSL modem to the WAN port of the router. Configure the router by connecting it to a PC with a cable and following the instructions in the user guide. Then afterwards, disconnect and connect through the wireless.
With both of these methods, each machine will likely receive a unique, private (only accessible on the LAN) IP address that is used to identify itself to the other computers on the LAN.
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