This is only supposed to give a short introduction to what you need, hardware-wise, to set up a simple Ethernet network. I only cover the two most common types of cabling, thin coax (10base2, BNC) and twisted pair (RJ-45, 10base-T and its faster variants).
Note that these two types of cabling are not mutually exclusive, since there are 10base-T hubs with an additional BNC connector.
One Ethernet card (NIC, Network Interface Connector) for each computer you want to network. Ethernet cards exist for nearly all bus types. Every computer can have a different card (though it might make your life easier if as many of them as possible have the same type), but of course they must have connectors for the type of cabling you want to use. Larger cards (ISA, NuBus, PCI) often have two or three different connectors.
Not all computers have expansion slots, and there are other types of NICs, too. Most PowerMacs and some PC motherboard have onboard Ethernet, usually with an RJ-45 connector. Older PowerMacs and most PowerBooks have an AAUI connector, you need an external transciever for this. There are also Ethernet adapters that can be plugged into the SCSI bus.
This is also known as Thin Ethernet, thin coax, BNC, or Cheapernet, though the last term is hardly used any more. It was cheaper than the olf 10base5 or Thick Ethernet, the original Ethernet developed by Xerox in the early 70s, which was more reliable, more powerful, but far more difficult to set up.
10base2 is not bad for a home network. As of 2003, it can be considered outdated but still supported, parts are readily available both second-hand (cheaper) and new, important when you need a certain part fast. Cabling is less flexible but tidier than 10base-T, and it is easier to find cards for older computers. I have never seen an 8-bit ISA card with an RJ-45 connector, and NuBus cards are at least very rare.
As I said, ideally you should have one open and one grounded terminator. Open terminators have a plastic cap that is usually green and are far more common. Grounded terminators are all metal and have a short chain attached to them, a chain made out of small metal spheres like the chains used for the plug of your sink. To ground the terminator, you attach the chain to a metal part of the computer. This is easier said then done, but luckily many network cards have an AUI connector two (it looks like the joystick port on a PC or the monitor port on a Mac), you can attach the chain there. You'll know how when you see it.
While having terminators is absolutely necessary (as soon as you remove one, the network is down), having one grounded is not. But it improves speed and stability.
Because of this need for terminators, adding a new computer always results in downtime of the network. You can, however, remove a computer without interrupting the other connections by seperating the T-piece from its NIC. So you could, theoretically, provide "empty" T-pieces to add computers later. I do not know whether this decreases performance of the network.
This type of network, using twisted pair cables with RJ-45 connectors, is the most widely used nowadays. Twisted pair, unlike coax cabling, supports far higher network speed. 100base-T is already fairly common and Gigabit Ethernet not unheard-of.
10base-T uses a star topology, all the computers are plugged into a central hub. This has its disadvantages, but on the other hand you can easily add and remove computers, and a single broken cable does not bring the whole network down.
There are hubs with an extra BNC connector, giving you the best of both worlds. If you have older and newer computers on your network, one of these is strongly recommended.
If you want to connect only two computers, all you need is a crossover cable. The following list is for two or more computers.
If you only want to connect two computers and got yourself a crossover cable, you simply connect them with the cable. There is really nothing you could do wrong, except having the wrong type of cable. You need a crossover cable, a normal cable won't work here.
If you want to be able to connect more computers and got yourself a hub, plug in the hub, and connect the computers to the hub. Again, be sure to use the right type of cable. You need normal, "straight" cables now.