The most basic difference between hard and soft water is that hard water reacts with soap to form solid soap scum, and soft water does not form soap scum. The formation of soap scum on your skin is the reason that soap seems easier to rinse from your body when you shower with hard water.
Hardness in water is caused by the presence of calcium ions, magnesium ions, or both. As their concentrations increase, water becomes harder. The combined concentration of calcium and magnesium is frequently referred to as total hardness.
When hard water is boiled in the presence of carbonate (CO3-2) or bicarbonate (HCO3-) the calcium and/or magnesium ions react with the carbonate and bicarbonate ions to form solid calcium carbonate, magnesium carbonate, or both.
Much of the ground water in the United States is trickled through underground labyrinths of limestone (calcium carbonate) and has high concentrations of both calcium and carbonate. This is the type of water that really wreaks havoc on plumbing, especially water heaters. A large portion of the total hardness in this sort of water is composed of temporary hardness.
Temporary hardness is complex, because its concentration is a function of the concentration of carbonates in relation to their reaction with calcium in magnesium.
Suppose your water has 100 parts per million of total hardness before boiling and 60 ppm of total hardness after boiling. This means it has 40 ppm of temporary hardness. The good thing about temporary hardness is that it is easy to remove by boiling or through precipitation with lime (calcium hydroxide). Boiling and adding lime are two of the older methods used to soften water containing temporary hardness.
Permanent hardness is simply the hardness that is not removed by boiling. If your water is gypseous — that is it has passed through gypsum in the ground — it will contain calcium and sulfate. When gypseous water is boiled, very little hardness is lost, because calcium is not precipitated by sulfate.
Many brewers add calcium chloride to their brewing water, and this is another source of permanent hardness. The good thing about permanent hardness is that it is stable. This applies more to commercial brewers than to homebrewers, but if water is stored hot in large hot water tanks and contains temporary hardness, the concentration of calcium will change over time. This problem doesn’t happen with water in which most of the total hardness is composed of permanent hardness.
One trivial point worth noting is that distilled water adjusted with calcium sulfate still contains some temporary hardness because carbon dioxide from the atmosphere contributes a small amount of carbonate to water. This means that no teapot or bathtub is completely immune to calcium and magnesium deposits.
With modern water-treatment systems able to do just about anything to any sort of water, the difference between the two types of water is less significant than in the past. However, for homebrewers using local water without any major treatment, the difference between temporary and permanent hardness is important. A quick survey of traditional brewing centers reveals that the dark-beer cities have water dominant in temporary hardness and the cities known for light/pale beers are more skewed to the permanent-hardness side.