Compost is one aspect of gardening that I don't like to discuss: Telling someone how to make really good compost is like telling him how to raise his kids. Each gardener has his own method of making compost, and each will staunchly defend that method. Besides, like parenting, there's no definitive way to make compost.
Practically speaking, it's important to have a container of sorts, although I know one country gardener whose compost is just a large pile, and a healthy compost it is. The size, shape and material that go into the container are largely matters of choice, but one rule that seems to hold true is that the container should have plenty of air circulation. You also want the container to be fairly dog-, cat- and raccoon-proof.
A square enclosure of cinderblocks or horizontal boards (nailed to four square posts) serves well. You can build a round enclosure with snow fence or chicken wire, but you need a lot -- the only practical way is to use the end of a fence roll. A 55-gallon drum works well, but you have to punch holes in it for aeration. There are also compost bin kits available through seed catalogues. They tend to be more expensive, but they do work if you don't want to go through the hassle of building one.
A practical size is about 4 x 4 x 3 feet tall. Some gardeners build three bins -- one for new compost, one for aging compost and one for finished compost. Personally, I think this complicates the process unnecessarily, but many a fastidious gardener would disagree. A heavy lid over the bin discourages most animal pests and keeps the compost neat. The lid should allow rain and moisture in -- perhaps a screen or chicken or rat wire stapled to a two-by-four frame. Compost shouldn't dry out -- the moisture helps organic matter break down into good fertilizer.
Once you have built some sort of structure, you should set up a platform, six inches above ground, on which you'll start the compost. The platform makes for maximum air circulation, which hastens the decomposition process. Or you could lay boards on cinderblocks, allowing perhaps an inch or two of space between the slats, or, if you're as lazy as I am, put down a pile of brush -- broken twigs, kindling, whatever -- which will, at least for a while, allow plenty of air circulation at the base of the compost.
It's pretty much up to you to decide how to build the compost. Mine is absolutely the lazy-man's bin, but it seems to work. Two years ago, I added fall leaves and straw to the twig layer and then I put down about two dozen shovels of dirt, pretty good dirt. And since then I've simply been throwing in table scraps, garden clearings and anything organic that doesn't go directly into the garden for one reason or another.
There's a lot of literature concerning the proper turning and aging of compost, but I've found that leaving it alone works well, as long as you don't try to use any of it for about a month after you've added the last fresh (uncomposted) organic matter. This is where a second or third bin might be useful. I leave my compost alone all winter, adding to it throughout the cold weather. Toward spring, I stop putting in table scraps: I give them instead to the chickens. By about March, the compost, which really hasn't been turned at all, is ready to go onto the garden. It's such a rich mixture that each year I have different volunteer vegetables coming up out of it. Last year, I had early cherry tomatoes in my compost bin; this year spaghetti squash appeared.