While most wines are associated with a dry, almost bitter taste with very little sugar, there are some wines that specifically seek to have sweeter, more sugary tastes.

The most prestigious of these wines can be very hard to cultivate and produce resulting in astronomical prices.  But one taste of that sweet wine and you'll understand why people go crazy for it.

Some wines achieve sweetness by manipulating a type of rot.  As the grape clusters sit on the vine far later than normal, the grapes become susceptible to bacterial and fungal growths.  But don't let that gross you out!  The winemakers know that there is one specific fungus that actually makes the grapes deliciously overripe. 

It's very hard to do this.  You run the risk of losing a large amount of the crop to other less favorable fungi.  Also, you can get much smaller yields from plants being pushed to this level of maturity year after year.  Finally, not all the grapes rot at the same time or at the same rate.  The famous Château d'Yquem in the Sauternes region of France sends out harvesters to hand pick individual grape clusters that are fully rotten, passing through each row multiple times over the weeks of harvest.

Another way to make sweet wine is to wait for the first winter freeze.  When the water in the grapes freezes, a lot of it crystalizes on the surface leaving only a very concentrated, mature grape behind.  A large percentage of a grape's flesh is water (just like your body is largely constituted of water) and these freeze-dehydrated grapes will be very very concentrated.

People sometimes ask why you can't just add sugar to make wine sweeter. Well, the natural sugars in a grape are going to taste way different than the refined sugar you buy at a store. There's all sorts of aromatic qualities in a mature late harvest grape that you wouldn't get out of a regular grape covered in sugar.

Of course, some people do chapitalize (add sugar before fermentation), but this will hardly result in a quality dessert wine. There are other reasons to chapitalize like eventually reaching a higher alcohol content.

Some wines like blush roses are made by intentionally halting the fermentation before all the sugar is converted to alcohol. This is, again, a totally unrelated sort of sweetness. It can be yummy, but there's no comparison. But it makes a lighter wine more like fruit juice which can be a good gateway for people who don't enjoy big dry wines.