Cloudy ice is most often caused by dissolved gases–mainly nitrogen and oxygen– that come out of water during the freezing process. Tap water also contains dissolved minerals–generally calcium and magnesium–which can be present in the form of bicarbonates and/or as calcium and magnesium sulphate. We’ll make sure to cover water hardness soon — but the short of it is that the sum of magnesium and calcium concentrations in your water is a measure of how “hard” your water is. Hard water can lead to cloudy ice cubes.

Cloudy ice can also form due to hindered structure formation. As water freezes, a thin layer of ice begins to form on its outermost edges and grows inward–forming a crystal lattice structure.

Ice Crystal Lattice Structure (Source:http://www.benbest.com/cryonics/icecryst.gif)
However, other molecules in tap water (such as the impurities and gases we just mentioned) do not fit in the lattice structure. So, if the ice forms faster than the gas molecules can diffuse away, the gases get trapped. Micro-bubbles form as the concentration of gases and impurities in the freezing water becomes supersaturated. These small bubbles get caught in the ice, and voila–you’ve got cloudy ice.