From the moment that a bottle of wine is opened and the wine is exposed to oxygen, it begins to change. This process is referred to as "breathing". The technical term for the change that occurs in the wine when it is exposed to air is oxidation.
If you have ever cut an apple in half and left it out on the kitchen counter for a while, you may have noticed that it begins to turn brown. This is evidence of oxidation. Much the same way that metal oxidizes and turns to rust, fruit oxidizes and changes chemically. If you have had the opportunity to taste the apple when it is first cut open and then again once it has oxidized you would notice that the apple tastes different. I find that after a short oxidation period (maybe an hour or so), the apple will appear to taste sweeter and have lower acidity. If the apple started out too tart or acidic, the breathing period may make the apple taste better due to the lower level of acidity. If the apple started out ripe and sweet to begin with, the breathing period may not be necessary and may actually result in a degradation in quality. The exact same principals apply to wines with regard to their tannin, acidity, and "fruitiness".
The amount of time it may be necessary for a wine to breathe in order to reach its peak depends on the varietal, age and complexity of a particular wine, and perhaps most important, your personal preferences. Oxygen can soften the tannin and help tight, closed wines develop their fruit flavor and balance in a relatively short period of time. Breathing can shorten the ageing process to hours instead of years. If a wine gets too much oxygen, it will fatigue and the fruit will dissipate. Wines that are older will often be more sensitive to the breathing process (i.e. the fruit will dissipate quicker in older wines as they breathe). So the trick to letting wine breathe is finding the balance between softening of tannin and dissipation of fruit.
Of course there is an alternative, let the wine breathe in the glass! Pour the wine in your glass when it is first opened, taste it, wait a while, taste it again, and repeat. You should notice changes in the wine as it continues to breathe.
The breathing process can be sped up by decanting the wine or simply by swirling it in the glass.
Unfortunately, most wines do not come with instructions on whether or not to let the wine breathe prior to serving and for how long. Fortunately most modern, moderately priced wines are ready to drink right away and require very little, if any breathing time.