The network adaptor for the fat-PS2 is a plug in unit that fits in the back of your PS2, and has an RJ-45 port for an ethernet cable.
It looks like the picture in this article.
If the back of your PS2 doesn't look like that, then you have 2 options:
1) buy the adaptor (quite hard as they're not made anymore)
2) trade in Fat-PS2 for a new slimline model which has a built-in RJ-45 port.
There's a few of us here,
Sjaak68,
Famine and myself who organise LAN parties on a fairly regular basis, and are very familiar with the 6 Player LAN setup.
You will need:
6 x PS2 with RJ-45 port.
6 x controllers
6 x memory cards with the 30 favourite cars selected (you can only race cars in "Favourites" in LAN mode)
6 x TVs.
6 x ethernet (RJ-45) cables.
1 x 6 (or more) port ethernet switch.
Lots & lots of power outlets, especially if running with G25/DFP setups too!
Check our Local Gatherings forum for more details on setups, and what it should look like when you're ready to race!
Properly executed it should resemble something like this.