The primary ‘colour of wine’ comes from the skin of the grapes. The juice that comes from nearly every variety of wine grapes when pressed is white or clear in colour. This is true of red grape varietals as well as white wines.
The colour or pigments of red grapes are found in the skins of the grapes. In order to make a red wine from red grapes, it is necessary to leave the skins in contact with the juice during the fermentation process. When the skins are placed/ left in the fermenting 'must' (which gets up to a heat between 21-30°C), the pigments leech out of the red grape skins and colour the wine. When red grapes are pressed immediately and the skins are kept out, the colour of the wine remains white and is considered a 'Blanc de Noirs' - a white wine made from red grapes; (e.g. Champagne).

 

White wines do not usually have the skins left in the 'must' during fermentation. If the wine is being made from white grapes, there is no benefit to the colour and if the wine is being made from red grapes, the skin contact would give an undesirable red colour to the wine.
Rosé wines can be made with 'limited/short' skin contact (leaving the skins in the fermenting juice for only a short period of time) - this method can be unreliable in obtaining consistent tinting from tank to tank, so blending is required before bottling. Some Rosé wines around the world can be made by adding a specific amount of red wine to an already finished white wine.

Why is the 'colour of a wine' so important to inspect? The colour of a bottle of wine with respect to its age can be an important clue in determining if a wine has been made from quality fruit, or has aged well over time.
For example, if a one year old bottle of Sauvignon Blanc is already a dark amber colour when the bottle is first opened, this could signify that the wine has not been made correctly or that the closure was faulty causing the wine to age prematurely and not taste its best.
The same can be said for red wines, if a young bottle of Merlot or Pinot Noir is already a brick red or a brown colour when opened and poured into a glass, chances are that there was a problem with the bottle closure, the temperature of cellaring or exposure to sunlight - and it too will not be at its best.