As somebody who is somewhat proud of his Hungarian background, when it comes to discussing wine, I always pretend to know how Tokaji Aszu, the most famous Hungarian wine is made. I realized however that I didn't really know much about it -- until I stumbled into the subject in Harold McGee's excellent book "On Food and Cooking - The Science and Lore of the Kitchen". So here it is.
The key point is that the grapes are allowed to become infected with 'noble rot', the mold Botryitis cinerea, that is a destructive disease of grapes and other fruits. "It becomes noble only in the right climatic conditions, when the initial infection in humid weather is followed by a dry period that limits the infection." The mold "dehydrates the grapes, concentrates their sugars, and transforms their flavor and consistency". The trick was invented in the Tokaj region of Hungary around 1650 (not 'Tokaji region' as McGee writes -- the '-i' at the end of the word in Hungarian means 'from', that is, Tokaji = from Tokaj). By 1750, it was adopted by the Germans; the French started using the method around 1800 in the Sauternes region of Bordeaux. [Hmmm... The French learning winemaking tricks from the Hungarians...]
The sweetness of the Tokaji largely depends on how many hods of 'aszu' grapes (that is, grapes affected by noble rot) was added to each 136 liter barrel of must made from 'normal' grapes; the traditional hods that carry ~20 kg of grapes are called "puttony" in Hungarian, and you can find Tokaji Aszu that is 3, 4, 5, or 6 'puttonyos'.
The terroir of the Tokaj region of course is just right for the aszu wine; this means south facing slopes with soils developed on volcanic rocks (andesites if I remember well) . In fact, the Tokaj Mountains are one of the oldest gold and silver-based mining districts of Tertiary-Quaternary volcanic arc of the Carpathians.
Here is more if you are interested. This might be too much detail, but looks great. And here is everything you wanted to know about the geology of the Tokaj Mountains.