NIC - Network Interface Card.
ok here is my setup, just look at it and you would be able to apply it to your comps.
1st computer:
2 NICs: 1 connects to the internet, 2 connects to the second computer
2nd computer:
1 NIC: connects to the first computer
Your case:
You can just buy 2 NICs, put 1 of them in the 1st computer, and the 2nd NIC in the 2nd computer and connect them with cross-over cable. A cross-over cable is not your average network cable. It has some wires crossed (as you can tell by the name of it

).
Whats cheaper? Well, you can look at the prices of NICs in your area, and basically add up the prices of 2 NICs and a Cross-over cable, and see what it will cost you. A good NIC to use can be with the speed of 10/100 (it will say so on the box). Thats what I use, and I can send a 87MB file from one computer to the other in 33 secs

.
If the prices of the kit and just 2NICs with a cross-over cable will be close, please provide us with a link (if possible) to both of the products, and we'll tell you which one is better.
I don't know which is the easiest to set up, because I have used only one configuration so far, but I can say it was pretty easy.