The most common method of producing a sweet wine is allowing the grapes to overripen, thus concentrating sugar and flavors.
Botrytis (or Noble Rot)
Botrytis is a fungus that causes grapes to rot, shrivel up, and dry out. Although this sounds (and looks) very unappealing, this is how the world's greatest dessert wines come into being. These are a step beyond mere "Late Harvest" in terms of time on the vine, sweetness, flavor complexity, and, typically, cost.
These rare dessert wines are made from frozen grapes. True examples are made from grapes left on the vines until midwinter, while inexpensive examples are often made from artificially frozen grapes. In both cases, water is trapped in the grape as ice and the resulting wine is highly concentrated. True ice wine takes the idea of late harvest to a whole new level. It is difficult to produce and consequently typically quite expensive. A specialty of Germany and, more recently, Canada.
These dessert wines are made by adding brandy during fermentation, which keeps the grape's natural sugars intact while raising alcohol percentage. These can be relatively light (Muscat de Beaumes-de-Venise) or quite heady (Australian Muscat).
These are made with grapes that have been laid out or hung up to dry for several months, concentrating their flavors and imparting a raisiny, dried fruit, nutty quality to the finished wine. Not as common as other methods but the specialty of certain regions, especially Tuscany where Vin Santo is produced.