The decanting of red wines has a long tradition in high quality wine service and can be done solely to add a special flourish at a meal. In decanting, the wine is simply poured into another container, usually one of clear glass or crystal. If sediment is expected, the use of a candle to assist in visualization adds even more to the ceremony. From the perspective of modifying the taste or appearance of a wine, the decision about whether or not to decant is based largely on two criteria, although the amount of published literature on the topic is very limited. (Terms that are not based on measurements, but are descriptive terms conventionally used by wine drinkers appear in quotations.)
Some young red wines--between three and 10 years older than the vintage date--can be harsh or astringent if consumed directly after opening the bottle. These are usually expensive wines that cost more than $20 in the U.S. market today and are produced with cellar aging in mind. Such wines have this harsh character because red wine is maintained in a relatively oxygen-free environment during aging in a bottle. Over time this environment results in a closed character for the beverages that is derived from the accumulation of particular aroma compounds. A wine's aroma will change during the first 10 to 30 minutes the bottle is open. Decanting accelerates the breathing process, which increases the wine's aromas from natural fruit and oak, by allowing a few volatile substances to evaporate. Decanting also apparently softens the taste of the tannins that cause harshness and astringency in young wines.
For optimal effect, a wide-bottomed decanter that gives maximum air exposure to the wine should be utilized. It is interesting to note, however, that chemists have not observed changes to these tannins after decanting. Less dramatic changes can be achieved by just uncorking a bottle 15 to 60 minutes prior to pouring. Keep in mind that many inexpensive to moderately-priced wines, as well as some more expensive wines, are intended for immediate consumption and will probably not improve with aging or decanting.
In older red wines the tannin reactions have proceeded long enough to reduce astringency. As a result, the taste is not as harsh but a sediment or precipitate may have formed in the bottle. This sediment is safe to consume, but if it is not removed it will make the wine look cloudy and taste gritty. Decanting leaves the sediment behind, yielding clean wine. (When decanting to remove sediment, a narrow container should be used instead of a wide-bottomed one.) In the case of older wines, one should not wait to pour the wine after decanting, but instead serve it immediately. The bottle bouquet of old wines, especially very old wines, can be exceptionally fleeting, often disappearing in less than 20 minutes.
In comparison to reds, white wines have little tannin and are not aged in bottles very long before serving. Thus they have little opportunity to develop bottle aromas that need evaporation. Instead their natural fruit aromas more specifically define their taste. Because these aromas are volatile, decanting actually results in a wine with much less of the aroma than the winemaker intended. In addition, because white wines contain fewer tannins and pigments, they don't produce the same quantity of sediments that red wines do.
The author would like to thank Kay Bogart for her help in preparing this answer.