To hook up two computers straight to each other, you need a special kind of cable called a crossover cable. It differs from regular cat5 ethernet cable, and you'll probably have to ask for it at you local computer place. The easiest and most efficient way in the long run is to simply have a hub or switch that you both plug regular cat5 cables into, because thay way more than 2 computers can be on the network at a time. Don't say that it'll never happen, because when I first got into networking computers years ago, I bought 4-port hub thinking I'd never need more than that, and now I have 4 computers and a router/firewall all going into the same hub, so needless to say I replaced that some time ago.
Anyway, crossover cable straight from machine to machine if this is a one-time thing, but both computers plugging into a hub with regular cat5 is pretty cheap depending on the hub you get, and regular cat5 cable is cheaper and easier to find.