As DataJunkie mentions above, Apfelwein is probably one of the easiest drinks to ferment yourself. All you need is some suitable apple juice or cider, yeast, and a clean container. Under the right conditions, you can have your own apfelwein in a little over a month. Here's how:
When starting apfelwein, the most important ingredient is your juice. Not all store-bought juice can be used because many manufacturers add preservatives to their juice to give it a longer shelf life. Since fermentation is just a fancy word for spoilage, you want to avoid preservatives as they'll keep the yeast from doing their work. Try to get 100% apple juice with no preservatives added—the only common preservative that won't block fermentation is ascorbic acid (avoid any sorbate like the plague). Apple cider, especially if you get it from a local cider mill, is excellent for apfelwein. Bear in mind that if you start with cloudy juice you'll probably end up with cloudy apfelwein: if you're the type that is uncomfortable with drinking something with a little yeast or other sediment in the mix, this is probably not for you.
Once you have your juice you can add sugar if you desire. This will have the end effect of raising the final alcohol content of your apfelwein. Don't go nuts with this, one pound of sugar will generally increase the alcohol content of your final drink by about 6% per gallon (i.e. if you add one pound to a 3 gallon batch it will increase the ABV by 2%). My favorite recipe calls for 2 pounds of sugar per 5 gallons of juice. Make sure the sugar is completely dissolved to ensure ideal fermentation.
You will also need an airtight container that can still allow the excess carbon dioxide to escape during fermentation. This keeps bacteria in the atmosphere from getting into your juice and turning it to vinegar while making sure the container doesn't explode from excess pressure. If you're going all out, you can buy a glass container called a carboy (also called a demijohn in some regions), a rubber stopper for the neck, and a plastic fermentation lock; a common one-way valve. If you're cheap you can take an empty plastic two liter (or other container) and make your own fermentation lock by taking a rubber balloon and poking a hole in it using a pin—the elasticity of the rubber will keep the hole closed until the balloon reaches a certain pressure, releasing the gas but preventing outside air from re-entering.
Getting it Started
Before you pour your juice in the bottle, you need to make sure it's sterile. This will prevent molds and other bugs from infecting your apfelwein. Iodophor and Starsan are standard brewers chemicals available in some specialty shops but household bleach will work just fine—make sure it's unscented bleach though, unless you like your apfelwein tasting like lavender soap (try not to use dish soap for the same reason). Rinse the inside of your container with a diluted solution of bleach—1 tablespoon per gallon or water should be enough—let sit for one minute, then rinse with hot water until the bleach smell is gone.
Now, fill the container with your apple juice/cider and add the yeast packet if you're using brewing yeasts. If your juice is unpasteurized, you can throw it in as is; the yeast on the skins of the apples when they're pressed should take care of the job in a few weeks. If you bought pasteurized cider or store-bought apple juice, you're going to need to add yeast—I recommend a good wine or champagne yeast that ferments to dryness. You can buy yeast from shops that sell homebrewing supplies (generally the same specialty shops that sell microbrews and other specialty beers) and from online stores or you can simply leave your juice out on the counter for a few days. I'd advise against the second strategy because it will attract fruit flies which carry bacteria which will turn your apple juice into apple vinegar quickly.
Once you've sterilized your container and thrown the juice and yeast in, attach the fermentation lock and put it in a dark, room temperature area for a few weeks. You should see some activity inside within 3 days depending on the temperature. Yeast generally ferment best in the 60s and low 70s (about 15-22 Celsius), much higher and they start producing hangover-producing fusel alcohols, lower and they go dormant, stopping fermentation. Cooler temperatures (around 14C as DataJunkie suggests above) will preserve more of the apple flavor while higher temperatures will make it ferment more quickly. At 70 degrees, the yeast will be done fermenting in around 4-6 weeks. In a general sense you can tell when it's done by when the fermentation lock looks to have been still for a few days (though fermentation may still be going on so it's best to leave it alone for at least a week after you last see activity). Most apple juices will ferment to around 5.5% to 8% alcohol by volume on their own depending on the initial sugar content. Obviously it will have more alcohol if you added sugar at the beginning. By comparison, most commercial American beers tend to be around 4-5% abv. If you're drinking it immediately, you're done! If you want, you can let the apfelwein age for a few more weeks or months to allow to yeast to settle out and the flavors to develop.
Some people like to carbonate their apfelwein (making it similar to champagne). I'd suggest you only try this if you're using brewing yeasts (as opposed to fermenting from unpasteurized juice) because it takes several more weeks and increases the risk of your apfelwein turning to vinegar. Also, there is a danger of your bottles exploding (similar to a dry-ice bomb) if you add too much sugar at this step so make sure you don't add too much and keep your bottles in a safe place where they won't do too much damage while they're carbonating. To carbonate, siphon or gently pour your apfelwein into another container (to get it off the yeast that has settled to the bottom of your original fermentation vessel), boil about 1 oz of table sugar per gallon (a little more than 1/8 of a cup or 28 grams) in water, then add it to your apfelwein. It's important not to splash your apfelwein too much because that will cause all those wonderful flavor-producing compounds to oxidize. You should also make sure the container you use to carbonate was designed to hold pressure (i.e. a soda or beer bottle), do not use plastic water bottles. Place the bottles in a warm and dark location for at least 3 more weeks, then chill them in a refrigerator for several hours before serving to dissolve the CO2
There are also those who add different ingredients though at that point it becomes a point of debate over whether it is apfelwein or not. Some add brown sugar or honey for a different flavor profile or add mulling spices and vanilla. Others add raisins after the initial fermentation (2-3 weeks after starting) to give the apfelwein a firmer body and mouthfeel.
Taste and Enjoyment
A warning; this stuff goes down like water and will knock you straight on your ass if you're not careful. It tastes like a fairly tart and dry wine with an apple aroma and aftertaste. Like all alcoholic beverages, that taste changes with age, the apple flavor coming to the forefront while the alcohol and yeast flavors mellow. As mentioned above, it's common to mix it with soda water or a soft drink such as Sprite to add a little sweetness if that's your preference.