Let's look at how dessert wines are made first. Grapes are sweet to begin with. Have you ever tasted a wine grape, just off the vine, around harvest time? Extremely sweet. They don't taste anything like those table grapes you nibble on at snack time. Matter of fact, wine grapes are said to have the highest sugar content of any fruit, and that's what makes them so great for wine. You need sugar, to act as food for the yeast, which eventually converts into alcohol.
Sweet wines can be made a number of ways. There are fortified dessert wines (such as Port and Madiera), where the grape juice begins to ferment, but is stopped short of full fermentation (where all the sugar is converted to alcohol, and the wine is dry). Fortified refers to the addition of brandy (see my blog on Cognac) into the fermenting grape juice, basically killing the yeast, and stopping any further fermentation. What is left is a sweet juice, fortified with a high alcohol brandy.
There are some "lesser" methods for producing sweet wines, which include the addition of sugar (known as chaptalization). The sugar can be in any form: honey, beet sugar, or in some cases, a concentration of grape sugars (known as Süssreserve).
When serving these sweet wines, keep in mind that, in general, you want your wine to be sweeter than the dessert you are serving. Usually these wines are so sweet they can be a dessert by themselves. Whites are typically served slightly chilled, and reds at room temperature. There are some classic pairings that you may want to try, and not all include dessert:
Vin Santo paired with almond biscotti (Cantucci)
Port paired with Stilton Cheese and walnuts
Brechetto d'Acqui - While not traditionally thought of as a dessert wine, I find this sweet, frizzy wine, to be one of the best wines to pair with chocolate. I know others think of Port, but try this one if you can find it.
Madeira (Bual or Malmsey) with anything you would add caramel to: apple pie, pecan pie