|Above left: corn sugar. Right: brown sugar. Below: table sugar.|
Corn sugar: This is a fermentable sugar most often used to prime beer before bottling. It's generally comes ground ultra fine so that it goes into solution easily. It can also be used to raise the alcohol of a wine or beer without altering the flavor. It is a very refined sugar.
Table sugar: This is sucrose, which yeast will ferment. It is by far the cheapest sugar for making wine, and can also be added to your wine or beer to boost the alcohol content without altering the taste.
Brown sugar: Brown sugar will, of course, also boost your alcohol content, but it will also change the taste of your wine. During fermentation and aging the brown sugar (which has some of the molasses left on it, as it is less refined than table sugar) imparts caramel aromas and flavors to the wine. Thus, you wouldn't want to use this with peach wine or a grape wine, but it makes for a very warming and cheery apple cider, and can even add nice complexities to a mead or beer.
Candi sugar: Also called inverted sugar. This sugar is often used in European beers, especially Beligian beers. It brings up the specific gravity, and therefore alcohol content, without much altering the flavor of the beer or wine.
Molasses: Molasses should probably be avoided. It will greatly affect the taste of your wine, and often not for the better. While it may bring some notes of caramel out, molasses is full of impurities from refining table sugar, and those impurities can lead to off odors and flavors in your wine.
Honey: If you've tried mead, you know the flavor that fermented honey imparts, and it can be very nice and refreshing, but it is a flavor all its own. Honey compliments many beers and fruits, and when used with fruit juice, a wine called melomel.
Malt extract: This is dried malt extract, and is often used in beers, but can on occasion add a grainy base to fruit wines if desired. Its use as such is not especially common.
Turbinado sugar: This is a washed brown sugar, leaving a little molasses, but not much. It imparts nice caramel hints, without overpowering the flavor of your wine, an excellent choice for sugar.
So hopefully this helps to answer, What kind of sugar should I use in my wine?