Kabel:
http://help.emule.nl/netwerkkabel/When you connect computers together it counts as a network, even when it is using a crossover cable.
All computers (hosts) should be given an IP address and a netmask. Unless you're sharing an internet connection, the gateway isn't needed.
Not sure about your eee, but on ubuntu you need to look at the panel - top right - where there is an icon for your network card. Right click and change the properties so that IP is set to something like "192.168.0.2", with netmask set to 255.255.255.0 - using this means you will communicate with all devices with IPs that start off "192.168.0."
The eeeepc should have same netmask but set its IP address to something slightly different, say "192.168.0.3".
Once you have set these up you should be able to open a terminal in ubuntu and type "ping 192.168.0.3" (address we used for the eee). It should start filling lines on the screen telling you how long it took to recieve a reply.
Try that for size and if it works we can move on to getting them doing useful stuff.
NOTE: This isn't linux's long way of doing things, you would have to do it in windows. A lot of people network their home pcs using a router, which has an inbuilt program that tells each computer which numbers to use (DHCP Server). This is when you would use the "automatic settings"