The whole concept of letting wine breathe, or aerate, is simply maximizing your wine's exposure to the surrounding air. By allowing wine to mix and mingle with air, the wine will typically warm up and the wine's aromas will open up, the flavor profile will soften and mellow out a bit and the overall flavor characteristics should improve.
Which Wines Need to Breathe
Typically red wines are the ones to benefit most from breathing before serving. However, there are select whites that will also improve with a little air exposure. In general, most wines will improve with as little as 15-20 minutes of air time. However, if the wine is young with high tannin levels, it will need more time to aerate before enjoying. For example, a young Cabernet Sauvignon will likely require around an hour for proper aeration and flavor softening to take place. Not that you cannot drink it as soon as it is uncorked, but to put its best foot forward give it more time to breathe. Mature wines (8+ years) are another story all together. These wines will benefit most from decanting and then will only have a small window of aeration opportunity before the flavor profiles begin to deteriorate.
How to Let Your Wine Breathe
Some erroneously believe that merely uncorking a bottle of wine and allowing it to sit for a bit is all it takes to aerate. This method is futile, as there is simply not enough room (read: surface area) at the top of the bottle to permit adequate amounts of air to make contact with the wine. So what's a Wine Lover to do? You have two options: Decanter or Wine Glass
Decanter - use a decanter,a flower vase, an orange juice pitcher, whatever - any large liquid container with a wide opening at the top to pour your bottle of wine into. The increased surface area is the key to allowing more air to make contact with your wine. Keep this in mind while setting up proper "breathing" techniques for your favorite wine.
The Wine Glass - Pour your wine into wine glasses and let it aerate in situ. This is certainly the low-maintenance method and typically works quite well. Just be sure to keep the glass away from the kitchen commotion, while it breathes in peace. * Tip, for pouring wine into glasses make sure that you pour into the center of the glass with a good 6-10 inches of "fall" from bottle to glass to allow for further aeration during the actual pour.
In general, the Aeration Rule of Thumb: the more tannins a wine has the more time it will need to aerate. Lighter-bodied red wines (Pinot Noir for example) that have lower tannin levels, will need little if any time to breathe.