When a wine is very sweet or contains a good deal of sugar it crosses the line into the territory of a dessert wine. At their best, dessert wines are well balanced with acidity to keep them from being too sweet. Ideally, a good dessert wine should be more sweet/tart than sweet. The trick to making a dessert wine is making sure that all of the sugar is not converted by the yeast, there are a few ways to accomplish this.
• Late Harvest
• Picking late gives you more sugar to water ratio in the grape
• The Noble Rot
• Freeze the grapes, remove the ice
• Think raisins
• Adding alcohol stops the fermentation
• Cold Stabilization
• Refrigeration stops the fermentation
• Chemical Stabilization
• Sulfur Dioxide stops the fermentation
Most dessert wines start with being picked late, even if another method is used in addition to ensure the resulting wine has residual sugar. The longer the grape matures, the sweeter it gets. The amount of sugar in a grape is in inverse proportion to the amount of acidity (just like all fruits, the riper it gets the lest tart it becomes). This means that the a grape picked too late may not have the acidity to balance the sweetness. The wine has so much natural sugar that the yeast dies out before it can convert all of the sugar. The yeast is usually chosen specifically for this task. In some cases tartaric acid (grape acid) is added to the wine to help the balance. This is not always a desirable or even legal thing to do. Some late harvested wines, notably in Alsace, France, are vinified completely or mostly dry (all the sugar is converted) resulting in a wine with little sweetness, but great intensity.
Botrytis Cinerea is a mold that can devastate a vineyard. A great deal of effort is taken to ensure this doesn't happen, unless you are trying to end up with a dessert wine. Botrytis is referred to as Noble Rot when you want it (and less pleasant names when it is not desired). This mold attacks the grapes, causing microscopic lesions through which it extracts water. The grape shrivels and what is left over is the solids, primarily sugar, acid and some trace minerals. The grapes are picked when they are just at the right level of sugar and the wine is made in the same way as any late harvest. It is not uncommon in the Sauternes region of France to pick the grapes individually over a period of a week or more, to get just the very best berries. In Germany and elsewhere this sorting is done on a table, where only the best individual berries are selected. The great Tokaji wine of Hungary is made by adding a paste of these botrytised grapes to the wine. The more baskets of paste (puttonyos) added, from 1 to 6, the sweeter and more intense the wine.
The incredible Ice Wines of Germany, Austria and Canada are created when the grapes freeze, allowing the water to be removed in the form of ice. While the best styles of this wine are from grapes that froze on the vine, modern technology has introduced quick freezers for duplicating the substance, if not the exact result of Ice Wines.
Placing the grapes onto straw mats to dry them, before making the wine is common in Italy. There the Recioto style is used most notably to turn Valpolicella into the more intense Amarone wine. The sweeter styles are still called Recioto while Amarone is the drier style. In France the technique results in wines referred to as Vin Paille. It is also used to make the Straw Wines of Germany and elsewhere. This ancient technique is just one more way to reduce the ratio of water to sugar in the grape. The wines made in this technique often have the same type of darker flavors you find in raisins.
There is a great deal more detail on the process here, but the simple answer is that alcohol is added to kill the yeast before they can convert all of the sugars. This is how Port, Sherry and other fortified wines are made.
Usually reserved for lightly sweet wines, it is possible to stop the yeast from finishing their jobs by simply refrigerating the wine. In this case, as with most of the other techniques a careful filtering ensures the fermentation process does not start back up. The term is more commonly used for the process of removing precipitates from dry white wines.
It is possible, but not altogether common, to use Sulfur Dioxide to halt the vinification process, resulting in a wine with residual sugar. Because of the possibility of off odors this is not really considered an ideal technique.