This picture is of a batch of my own hand-crafted tempeh and here's how to make your own tempeh. But first...
What is Tempeh?
If you are unfamiliar with tempeh, I'll just say that it is a fermented soybean product. For more, read up here. It's related, but quite different from tofu. While the tofu process is similar to cheese (the curds and whey thing), tempeh is fermented and is, I guess, more analogous to yogurt.
Well, lots of reasons. If you are a vegan, B-12 is hard to get, since it's mostly found in meat, dairy, and eggs. B-12 is needed for red blood cell production and so vegans can be susceptible to anemia. While some caution is advised against relying on tempeh for B12, since levels vary between batches, tempeh is the richest vegetarian source of B12. People also like it because it has a more "meaty" texture than tofu and can be used more as a meat substitute in recipes for variety. There are many health benefits to tempeh as with all soy products (although if you deep fry it in palm oil it's kind of a wash as far as your heart is concerned). For example, tempeh is cholesterol-free. Finally, for those of us who don't keep a vegetarian diet, it's simply economical to eat tempeh every once in a while, as it's cheaper than a lean steak. And if you needed another reason to try it, tempeh is delicious. Don't assume that if you don't like tofu, you won't like tempeh; they are very different.
2 1/2 c. soybeans
2 Tablespoons vinegar
1 teaspoon Tempeh starter
Process: My instructions presume starting with split, hulled beans, which can be hard to find. Starting with whole beans requires rubbing them between your hands after they have boiled (and cooled) in order to break them in half and get the hulls off (which, fortunately, float in water). Also, there are many variations on this process. I am just writing up what worked for me (though I summarize other suggestions along the way).
Cook soybeans for 1 hour at a boil, skimming off any skins that float to the top (if some are left, that's OK). Drain and pat beans dry in an absorbent towel until they are surface dry. Put beans in dry bowl. Having beans too wet is one of the most common causes of a bad batch.
When the beans are cooler than skin temperature, add vinegar. Mix well. Sprinkle beans with tempeh starter. Mix well for 1 minute to distribute evenly.
Put into 2 zip lock bags or other containers and incubate for 24-48 hours at 28-33C degrees (82.4-91.4F) (optimally 30-32C (86-89.6F)) degrees. Basically, keep it as close at you can to 88F/31C as possible. Be aware that at about the 12 hour mark, the tempeh will start to generate its own heat and you'd need to turn down your source of heat.
 What to use for containers?
Plastic Ziploc bags (7"by 8"), perforated with holes at distance of 1 cm by a .6 mm diameter needle/nail.
Tetra Brik (disadvantage: not transparent)
Plastic sandwich Tupperware
Go with the original: banana leaves
How to incubate?
If you live in a warm climate, experiment with leaving it outside. Otherwise, the old fridge or Styrofoam box method. Or try your oven using only the light bulb. Or rig a space heater to a thermostat. Or, if you really want lots of control and accuracy, go and spend the big bucks on a proper chicken incubator with circulating fan and thermostat.
Temperature / Time?
30-32C for 36-48 Hours (tempeh.info)
88F (31.1C) for 24 Hours (veggiemenu)
?C for 24-36 Hours (ellenskitchen)
88F (never less than 85 (29.4C) or more than 95 (35C) for 24-30 Hours (motherearthnews)
30-32 (86-89 F) for 24 hours minimum (henrynugroho)
88F (31C) for 24-36 hours (Manfred).
88F for 24-36 hours (pvachuska)
I first got the free sample from the Europeans and then for this batch I went with The Farm.
Finally, if your first batch doesn't work out, do not get discouraged. My first batch was a total failure (I think due to my incubation method, the oven with an incandescent bulb wasn't keeping a constant 88F). These pictures are the result of my second attempt, which went well.
The Book of Tempeh, 160 pages, by William Shurtleff and Akiko Aoyagi
This is the Tempeh Bible. If you are going to make some tempeh at home, you absolutely must buy this book (or find it in your local library). It is wonderful for nutritional, cultural, and historical information. It also has info on variants of soy tempeh and how to make your own starter plus lots of recipes, including traditional Indonesian dishes and creative uses for working tempeh into Western classics. It also has a directory of tempeh vendors around the world, though the book (my copy at least) is very old and possibly out of date in this regard.
I also have The Tempeh Cookbook by Dorothy Bates which is mostly recipes, but has some good info about making it. There are probably better books out there.