People often talk about "dry wine" as something markedly special. You'll sometimes hear people use the word "dry" in reference to wine as a synonym for excellent. In an informal survey of academics at a party where all of them were drinking wine, and considered themselves fairly knowledgeable about wine, three people told me that a dry wine was an aged wine; two people told me dry wine was put on the labels of high quality white wines.
They were wrong, of course. "Dry," for wine, or beer, refers to the fact that the beverage is comparatively low in sugar because fermentation has converted the natural sugar of the grapes into alcohol and carbon dioxide. Sometimes the process is referred to as "fermenting out" the sugar. The naturally occurring yeast that causes fermenting has consumed the sugar and converted it to alcohol, carbon dioxide, and heat. The wine master ordinarily stops the fermentation before the sugar is consumed, but in order to make a dry wine, the yeast is allowed to do what it does best. In Europe the law stipulates that in order to be labeled "dry" the final wine is .4% or less (that's 4 grams per liter).
In order to make non-dry or "sweet" wine, the winery may add sweet grape juice (or juice concentrate) back to a dry wine; this is typically done in order to increase the alcohol content, rather than to make it taste sweeter. In California, winemakers are legally prohibited from adding sugar to wine. Other ways include adding additional alcohol to the wine, which is how Port or other "fortified" wines are made, or, most commonly, by stopping the natural fermentation process by adding sulfur to the wine, or cooling it to the point that the yeast can't survive, or a combination of both. Then of course, with the right sorts of wine, bottle aging tends to increase the dryness of a wine.
In English, you'll notice wine labels with "dry" or sometimes "Extra Dry" as a description; you may note sec, trocken, secco, or asciutto, on imported wines. The very dry wines often have a lovely character, but they can take a bit of getting used to if you typically drink Rieslings or other, sweeter wines. Keep in mind that most red table wines are dry by nature, even if they lack a label to that effect. Your Shiraz, Merlot, Cabernet Sauvignon are likely dry.
You don't have to make the transition all at once, though. The Chateau Ste Michelle Dry Riesling, for example, retains much of the character and robustness of their traditional Riesling, just without quite so much sweetness. This makes it awfully nice with food, especially zesty, spicy, rich, or salty dishes. And gives you a chance to experiment with a wine that's familiar, extremely drinkable, but a good deal less sweet.
Very dry wines are also excellent to cook with, for making fondue, marinades, dressings, and reductions, because they're very flavorful without becoming syrupy. The process of experimenting is half the fun of buying and consuming unfamiliar wine, of course.