Fuck Yeah Fluid Dynamics sums up the science of champagne bubbles:
During secondary fermentation of champagne, yeast in the wine consume sugars and excrete carbon dioxide gas, which dissolves in the liquid. Since the bottle containing the wine is corked, this increases the pressure inside the bottle, and this pressure is released when the cork is popped. Once champagne is in the glass, the dissolved carbon dioxide will form bubbles on flaws in the glass, which may be due to dust, scratches, or even intentional marks from manufacturing. These bubbles rise to the surface, expanding as they do so because the hydrodynamic pressure of the surrounding wine decreases with decreasing depth. At the surface, the bubbles burst, creating tiny crowns that collapse into Worthington jets, which can propel droplets upward to be felt by the drinker.
(Hat tip: Kasia Cieplak-Mayr von Baldegg)