Supersaturation
Carbonated water is a supersaturated solution of carbon dioxide gas in water.  It is stored at very high pressure in a bottle.  This lets the water dissolve more carbon dioxide gas than it can at normal pressure.  When the bottle is opened, the pressure falls to the normal atmospheric pressure, and the carbon dioxide gas escapes slowly.  Certain things such as small bubbles in the bottle can make this process faster.  Such bubbles can be caused by shaking the bottle.  Another solute such as sugar powder or a widget can also trigger precipitation.