Wine is a complex substance. Open a bottle of it now, and it may taste fine. Open a bottle of the same wine several years from now, and the wine becomes something else entirely. Rough edges in the flavor will have smoothed out and the overall flavor of the wine will be more balanced and mellow.
Why age wine?
Well, as stated above, the many components of the wine will mellow out over time, and bring a harmonious balance to the flavor of the wine. The tannins, present in red wine, bring a bitter and astringent flavor. Given time, the tannins help to age the wine. As wine ages, the tannins precipitate out of the wine, leaving a smoother and more mellow flavor. As the wine ages, the acids will lose a bit of their bite, allowing the fruit to come out more. As the components of the wine age, they also bring more complexity to the flavor. What you end up with after several years of proper aging is a complex and balanced wine.
What wines should be aged?
There are certain guidelines as to what wines should and should not be aged, but they are simply guidelines, not hard and fast rules. Most wines from California are meant to be consumed immediately, and should not be aged. Most white wines should not be aged, as they have a lower amount of tannins than red wine. It is the tannins that are present in red wine that allow the wine to age well. There are a few white wines, however that will improve with a couple of years of aging. Most wines under $25 don't need to be aged. Many French wines are meant to age and mature over several years. There are exceptions to these rules. Go into a wine shop, and ask which wines they would recommend for aging. They can recommend several good wines, along with some ideas of how long to age the wine.
How do I age wine?
Proper storage is essential for a wine to age well. Cooler temperatures slow down the aging process, allowing time for the complexity of the wine to develop. The wine should be stored at a constant temperature of 50-60 degrees F. Fluctuating temperatures will cause the wine and the cork to expand and contract, loosening the cork, and exposing the wine to oxygen. The wine should be stored on it's side, so that the wine is in contact with the cork, preventing it from drying out. The storage area should be dark, and have plenty of moisture to keep the cork from drying, which may loosen it, and subject the wine to oxidation.
Once you have a proper storage area for your wine, go to your local wine shop, and get their recommendations on a good aging wine. Buy a case of wine, and then open up one of the bottles when you get home and taste it. Put the rest of the bottles in storage, and start opening them after a couple of years, and note how the flavors and the complexity of the wine have changed. A new bottle of wine may be good, but a properly aged bottle is even better.