Every six months or so, I remember that I really like cider. Unfortunately, I don't drink much of it, because the cider industry is in a pretty terrible place right now, as it has been for decades. Traditional English cider (that's scrumpy, not Strongbow) is difficult to find in England, and impossible to find anywhere else. There are a few small American cideries doing good things (Alpenfire in WA, Farnum Hill in NH, and West County Cider in MA, to name a few), but they're quite expensive and have limited distribution. Basque cider is absolutely unique, but, once again, hard to find and expensive. Cider from Normandy—often considered the world's finest—is what I buy most often, but Cidre Dupont is the only kind I've been able to get my hands on.
Making cider at home can be a great alternative to the mass-produced, backsweetened swill that lines supermarket shelves. Of course, to do it properly, you need apples and a press—but I'm not that committed. So I start with juice.
During the apple harvest, many farms will sell freshly pressed juice, ideally from traditional cider apples. These varieties have higher levels of tannin and acid than eating apples, so they produce much more flavorful ciders. The rest of the year I buy cider from Whole Foods, because it comes in gallon-sized glass jugs, so you don't need another container to ferment it in. All you need is a stopper and an airlock from your local homebrew store; plus, you can re-use the jug to ferment other things. (Carlo Rossi jugs also work for this purpose.)
If you're using cloudy juice and wish to produce a clear cider, you'll want to add some powdered pectic enzyme. Add one teaspoon per gallon of juice prior to fermentation.
The first thing I had to do to make cider was forget everything I'd learned about making beer. When you make beer, it's important to aerate to promote yeast growth, because you want a quick, full fermentation. With cider, you want a long, slow fermentation, ideally leaving some residual sugar behind. This preserves the subtle flavors of the fruit. So when making cider, you should minimize aeration to slow the yeast down.
If your juice is unpasteurized and unsulfited, you can use the natural flora in the juice to ferment it. Otherwise, you'll want to pitch a small amount of yeast. If you're using dry yeast, I recommend 0.5 - 1 gram of yeast per gallon of juice—less than half the amount of yeast recommended for a beer of a similar strength. Fermenting at low temperatures will also help slow down the fermentation. 55˚ - 60˚F is a good target, but check the temperature range on the yeast you're using to make sure it can ferment at those temperatures.
You can use a variety of yeasts to make cider. For my first batch I used Danstar's Nottingham ale yeast, but it left strong yeasty flavors in the cider even after conditioning. Since then I've used Wyeast 3068, a German Hefeweizen yeast, and Wyeast 4766 cider yeast. Don't expect yeast to produce the same flavors in cider that they do in beer. I've used 3068 a number of times, and it produces very strong banana and clove aromas in beer, but in cider it produced neither. Both of these ciders were good, but I'd give the edge to 4766, because it enhanced the cider's acidity better. There are also a wide variety of wine yeasts that I'm planning on experimenting with.
Let the cider ferment and condition for 3-4 months before bottling. If you have a hydrometer, you can check the residual sweetness of your cider. A gravity under 1.002 qualifies as dry, 1.002 - 1.012 is off-dry, and over 1.012 is sweet. If you follow this method you won't make a sweet cider, but I have managed to produce stable off-dry ciders. My ciders have finished at 1.004 (Nottingham), 1.006 (4766), and 1.008 (3068). If you use a champagne yeast (Saccharomyces bayanus) or ferment at warmer temperatures, your cider may finish drier.
Good resources on cider making include:
http://www.cider.org.uk (and Andrew Lea's book Craft Cider Making)