Oxygen in general, and the oxygen of the air in particular play a fundamental role in the production and the evolution of wines. 

Oxygen and wine making

• During the alcoholic fermentation, the yeasts need oxygen. Open air racking and micro-oxygenation are commonly performed at this stage. 
• During the maturation of wines, oxygen also plays an important role, noticeably in wines matured in barrel. It interacts with several components of the wine, including with the anthocyanins and the tannins. 

To understand more about the role of oxygen in the wine making process, download this very complete article about micro-oxygenation. 

Aging of wines and oxidation

• The aging of a wine in bottle is traditionally referred to as a "moderate oxidation". Cork is naturally porous, so a bottle stopped with a natural cork will never be completely air-tight. In consequence, a small quantity of oxygen from the air slowly dissolves in the wine inside the bottle.  
• The oxidation of the wine in the bottles results in a series of reactions that alter the aspect and the taste of the wine. In a red wine, the partial oxidation of the ethanol (the alcohol of the wine) increases the precipitation of the anthocyanins. The oxidized ethanol forms bridges that glue all the anthocyanins  and the tannins together until they eventually become to heavy to float in the wine and end up forming  a black sediment at the bottom of the bottle. 
• The aromatic components of the wine are also affected by the presence of oxygen.