The degree of water hardness is directly proportional to the concentrations of mineral ions in the water.
Although other mineral ions are present, calcium & magnesium are the dominant chemical species found in solution. Hardness is usually expressed in terms of the amount of calcium carbonate (CaCO3) present in solution. Water hardness here will be expressed as parts per million (ppm). A classification of “soft” versus “hard” water is shown here (source):
The most immediate aspect of water hardness is that soft water has less buffering capacity, which means that pH can abruptly drop in an aquarium that has soft water. In aquariums with hard water, pH tends to be stabilized. (source)
Extremely soft water in aquariums can pose management problems. Some aquarists believe that distilled water or conditioned water (i.e.: Culligan) is pure & therefore good. This is not the case.
Fish lose minerals (electrolytes) through gills & urine; minerals must be replaced for optimal health.
High mortalities associated with freshwater fish kept in soft water can be attributed to mineral & electrolyte imbalances.
Total hardness affects the osmoregulatory balance between fish & its environment. Certain fishes have evolved in their “native” water supply which may be very hard or very soft. Other fishes can tolerate a wide range of hardness levels as they tend to change naturally in the environment due to natural fluctuations of water level, rainfall, & other factors.
Knowing the hardness of your water supply & your aquarium will help stores to recommend fishes that are best suited to your water, otherwise, you can adjust it as necessary.