Water at room temperature contains air (air is soluble in water). When cooling down water it can handle less and less air in solution. This air then gets expelled from the water. This normally isn't a problem, but when making perfect ice cubes this air will make horrible blemishes in the middle of your artwork.
\ / cold mould
Ice forms at the edges first, enclosing the water and isolating it from the atmosphere. When the water cools further more air gets expelled from the water and this creates bubbles under the ice
As the ice crystals grow, even more air is pushed out of the water until it finally freezes solid. As the air expulsion is directly related to the ice growth the ice contains minute bubbles throughout its entire volume1.
To avoid this industrial ice cube factories make use of cooling systems where the cubes are exposed to free flowing water. The water will carry away the expelled air before it is trapped under the ice.
Because the method also uses localised refrigerating elements, the water is never enclosed in ice. When the ice cube has matured into a nice block, the water is shut off and the mould is heated. The cube then slips and can be scooted away.
Szlater is kind enough to remind me that demineralized water will also do the trick. He also suggest that you boil the water before you freeze it: "Boiling the water drives out the dissolved gases... as the heat of the water increases its ability to dissolve gases goes down. Some gas might be reabsorbed during freezing, but this would be minimal. It's a much easier method to use at home."
1 - Except the first few millimetres, when the water is not yet saturated with air. The air that gets expelled from the ice gets eaten by the water until it is saturated.
I really really should mention something about dihydrogen monoxide's properties as an eutectic mix with 0.0038 wt% air and phase diagrams here, but come on - it's a weekend!