Years ago they may have been one or two different types of wine glasses. Since then, the market is flooded with a wine glass to go with a specific wine. Is that all necessary? Not really, but there are some basics about what glass to use with red or white wines. Not that the wrong glass will take away from tasting the wine, but the right glass may make it easier.
I am sure some of you older folks will remember the classic champagne glass.
These were the glasses that many newly married couples got as part of their wedding registry gifts. We still have a set that pretty much looks like this picture.
But to really appreciate champagne and how much effort the wine makers put into it, having a better shaped glass is so much better. 
This shape allows the drinker to watch as all the tiny bubbles slowly rise up the glass and an endless display.
Red wine glasses generally should have a bigger bowl.
This allows the drinker to put their noses right into the glass to appreciate the various bouquet/aromas that red wines have. This type of glass also allows you to swirl the glass more so that the aromas can be even more detectable, plus you can examine the legs of the wine. When you see these legs (tears) they are an indication of the alcohol content of the wine. The higher the alcohol, the large legs you will see. That should also represent that the wine will have a full body.
White wine glasses generally have a more of a U shape and more upright helping the aromas to be release and keeping the wine temperature more constant. 
Port and Sherry glasses should always be smaller than your Red or White wine glass. 
This helps direct these wines to the back of your mouth so that the sweetness does not overwhelm you
Cognac While Cognac and Brandy are not wines, they are distilled from grapes, not from grains like whiskeys are. I had a kit to take wine and to distill it into a brandy. It was a fun experiment, not resulting in any good brandy, but interesting. 
The glasses used for Cognac are different then those for other wines. They allow the drinker to swirl the cognac to release its enormous boquet which will aid in the actual tasting of the cognac. When I drink cognac, the first thing I do is put an ice cube into the glass, swirl it around a couple of times and then dump the ice cube out. Then I pour my cognac. Adding that little bit of water, really brings out more of the flavor in the cognac. Try it sometime.
However While the appropriate glass can enhance the pleasure in drinking wine, please do not refrain from drinking wine because you do not have the proper glass! I think the only vessel I do not like drinking wine out of is, a plastic cup! My wife feels real strong about this. She wants her wine glass! When we go out on picnics or other outdoor events, even boating, she always brings one of her wine glasses.