network newbie + linux newbie = migraine

The setup: I have a cable modem, a basic 4-port wired linksys router and two computers. One network adapter on each computer. Everything "works" fine except... I have a problem with the ip addresses being consumed.

While trying to learn how to use ssh I discovered the local ip addresses on both machines were changing daily, going up, up up. After a few days they were using 192.168.1.107 & 108 respectively. Decided to take the spare puter offline, do a fresh install of latest kanotix on my main machine here and take it one step at a time. Rebooted cable modem and reset router to default. Spare puter is still offline, nothing should be using 192.168.1.100 or 101 but this machine is now up to 192.168.1.102. Why? Even using dhcp (for local), the router should issue the same ip unless it is taken by something else, no?

There is a metric ton of info out there on dhcp, static ip addresses and the like but it is a total blowout overload greymatterbufferoverrun for me. Where do I start?