Think about this. You have a really nice bottle of wine that you want to open, but you look around and you do not have any cork screw! Now, I have see endless amount of Facebook pages on how to open a bottle of wine without a cork screw. About 90% of those never work and you wind up breaking the bottle.
Hence the Cork Screw! 
There are so many different types of corkscrews. You could go into any department store, wine retail shop or online shopping centers and find endless amounts of corkscrews. So many different types, sizes and shapes.
But before I even talk about the different cork screws, lets look at a little history. No one knows for sure who the inventor of the corkscrew was, but it is believed it was derived from a similar device that would remove bullets that were stuck in a rifle. However, the first known patent of the corkscrew dates back to 1795 by Samuel Henshall 
The top had a wooden handle, and a disc was added to prevent the from screwing too deep into the cork. This cork screw proved to be so useful, it was widely used for over a 100 years.
In 1882, Carl F.A. Wienke, filed a patent for the Waiters Friend. 
This cork screw became very popular especially in restaurants and bars. The lever action made it very easy to withdraw the most stubborn cork.
Another patent in 1888 by H.S. Heely was the double winged, or better known as the Winged corkscrew. 
This became the most popular cork screw in people’s homes. It was easy to use, hardly ever failed, and it was sort of magical as the cork came out of the bottle with hardly any effort.
Recently there have been so many different types of corkscrews put on the market. Some table top, some mounted on a wall, some using forced air to pus out the cork. They all work, it can just be a matter of taste and how you want to show your guests how you pull a cork.
The most unique and to me most important development in cork screws is the Coravin 
Most of us know that while oxygen can be a friend to wine, allowing it to breathe , helping reduce some of the harsh tannins in red wines. But it also can be an enemy in that the longer a wine removes in contact with oxygen, it starts to break down. This oxidation can ruin many a good wine. The Coravin was patented , approximately 2010, so that it does not have to extract the cork, but instead, inserting a small needle and allowing the pouring of the wine without compromising the wine itself. This has been a great way to allow the tasting of some of the more expensive wines in the world.
But back the basics that most people will be using in their homes. Whatever cork screw style you find comfortable for you, the most important part is called the worm. 
That is the part that penetrates the cork deep enough so that you can pull it out of the bottle. But the most important thing about the worm is that it needs to be smooth. Many of the early winged corkscrews basically had a auger for the worm. It was sharper edges. It penetrated the cork with ease, but unfortunately many times is ripped the cork, so out came the cork screw, but the cork remained in the bottled!
That always brought up another dilemma. How to get the cork out of a bottle, when the cork is broken or chewed up. The easiest way and I have seen countless people doing this, is to push the cork back into the bottle. This does work, you may lose a little wine, and you probably will have some pieces of cork floating in the wine. But you can carefully pour that into a carafe (which is an art in itself), or use something like cheese cloth to strain it.
Now, the formal way, but not always the easiest is using called
le-creuset-butlers-friend 
With this device, you push the two knife like blades onto either side of the cork. Then you gently start to twist and turn, slowly pulling upward until the cork is removed. Trust me, it takes some practice doing this. I had purchased an inexpensive type (less than $15) and discovered that it never worked. The knife blades were too thin. I did go out and buy one for about $35. Still it takes a bit getting to use it right, but it will work.
But then again, today we do have some many good wines that use the
metal screw top. Virtually impossible to fail while opening a bottle of wine!