Let it breathe
First things first, merely opening the bottle does not let the wine "breathe." This is one of the most common misconceptions about wine. The best and most entertaining way to accomplish real aeration is to pour the wine into a decanter/pitcher. It doesn't matter if it's fancy or plain, the purpose of the decanting is to get air into the wine.
If you're at home, get your wine to breathe by pulling the cork and pouring the wine into the pitcher from the greatest height you can achieve without spilling. The idea is to splash it as much as possible to maximize the wine's contact with air.
At a restaurant, answer the waiter by saying yes, please open the wine now. Then ask if she'll decant it for you. If the restaurant isn't set up for decanting, then ask that your wine be opened and glasses poured for everyone, even if you're saving the wine for the next course.
Your wine will do a lot more breathing in your glass than it ever would have while still cooped up in the bottle. Many young, concentrated red wines, like an expensive Cabernet Sauvignon or Bordeaux that can age for several years, undergo a sort of micro aging process by getting lots of air into them.
The same goes for a young and concentrated or especially astringent white wine, such as a white Burgundy. The reason is that forced exposure to air begins to oxidize a wine, causing subtle chemical changes that affect both the flavors and the texture.