The first stage is oak or barrel aging where the water, alcohol and acid present in wine extract chemicals, sugars and flavors from the wood.  This stage is anywhere from one week to 2 years long and the rate of extraction drops off hyperbolically with time.   The amount of flavors extracted in the first use of a barrel will never be reached again.  The amount of time required for a certain level of extraction for the third use of a barrel is twice the time required for the second use.  Wineries age part of a batch of wine in new barrels and the rest in old barrels to achieve the desired level of oak while minimizing cost and time.  In addition to extracting flavors, sugars and tannin from the oak, the wine is concentrated during barrel aging because water and alcohol evaporate through the porous wood.  Five percent, on average, of the wine evaporates per year of barrel aging so there is less wine available for sale.  Humidity and air circulation around the barrels affects the evaporation rates of water and alcohol.  If humidity is low, water evaporates faster than alcohol and the ethanol content of the wine increases.  If the humidity is high, then alcohol evaporates faster than water and the ethanol content drops.  The humidity required for constant alcohol content is about 68% relative humidity.  A very good, but highly technical discussion of barrel aging of wine is "Maturation of Wines and Spirits: Comparisons, Facts, and Hypotheses" Vernon L. Singleton,  Am. J. of Enol. Vitic., Vol 46, No. 1, 1995.  Dr. Singleton, who is now retired, dedicated his career at UC Davis to the study of how wine ages.

The second stage of aging is bottle aging where slow chemical reactions combine chemicals in the wine to produce new flavors that were not present when the wine was bottled.  Other chemicals combine and are too heavy to remain is suspension, so they settle to the bottom of the bottle as sediment.  Wines that are meant to be aged a long time have a high level of tannin when they are bottled from time in contact with the grapes skins and oak.  These tannins help preserve the wine, but over time attach themselves to chemicals in the wine and settle out.  Because of this, tannin is known as a finning agent.  Wines with high levels of tannin taste bitter (like chewing on a used tea bag) when they are young and become smoother or more mellow as the tannin level drops.  Some of the sharper/harsher flavors present in young wines are removed by the tannin and settle out as well.

The third stage of aging occurs once you open the bottle and expose the wine to air.  Some wines benefit from this more than others and some people decant wines into wide mouth carafes to "allow the wine to breathe."  The breathing process allows strong odors to bleed off and oxidizes the wine by bring it into contact with the oxygen present in air.  Too much breathing time or leaving wine in a half empty bottle for several days can ruin the wine through too much oxidation.