Some of this may be old news, but some members reading the thread may not have the same level of experience, so within reason I'll try to be detailed but concise.
GH is general or total hardness, and when it comes to this hobby that is primarily if not exclusively the measure of dissolved calcium and magnesium in the water. Most of us use municipal water, so there are strict controls on many substances like iron, copper, etc. But none on calcium and magnesium. Well water (private) is obviously different.
Water in its pure form does not exist in nature (except as condensed water in clouds, but that is not on the earth's surface); it is a powerful solvent, meaning a substance that easily dissolves other substances to create a solution. As rain falls, it picks up many gasses and particulate matter, and it continues to do this as it passes through the ground. Natural water values therefore vary with respect to hardness and pH because the water acquires specific properties from the landscape. Water flowing over or through rock will assimilate minerals from the rock, becoming what we term “hard” water. Water flowing through soils that contain organic matter will be “soft” because the organics bind with and thus remove minerals while creating acids that enter the water. The pH is largely the result of the hardness as well as the amount of carbon dioxide dissolved in the water.
As each freshwater fish species has evolved over thousands of years, their physiology has adjusted to the water values that occur in their respective habitat. We refer to these values as water parameters, and they include general and carbonate hardness, pH and temperature; each of these has an impact on fish. While many fish species appear to be somewhat adaptable, their physiology can be negatively affected if the parameters are outside the fish’s natural preference. They do not adapt as some unreliable sources will tell you, or if the species does "adapt" there are limits and other factors. Providing suitable water parameters in the aquarium is therefore an important aspect of providing an environment that is less stressful—and this directly relates to healthier fish.
Fish are directly impacted by GH [and TDS, total dissolved solids, which include GH but much more, and is outside the scope of your question]; their growth, the transfer of nutrients and waste products through cell membranes, spawning (sperm transfer, egg fertility or hatching), and the proper functioning of internal organs such as the kidneys can all be affected. Very generally (there are always exceptions), fish that have evolved in hard water must have calcium especially in the water they live in, in order for their metabolism and physiology to function well. Fish that occur in soft water do not need this, and any dissolved calcium in their water will be filtered out by the kidneys as a toxin, and here is where trouble occurs...the kidneys over time become blocked and the fish dies. The level of GH, the species, and the duration factor in. A study carried out in Germany found that the lifespan of cardinal tetras was directly related to the level of calcium (GH). The higher the GH, the shorter the life.
Of the four parameters, GH is the most important, followed perhaps by temperature (this drive the fish's metabolism) and then pH. KH is last. Again, this is very general, but it is a good guide. The GH, KH and pH are very closely connected, and the GH and KH will impact the pH. Which is why it is pointless and extremely dangerous to attempt adjusting the pH without including the GH and KH. i won't get into this here.
You mention water softeners...depending how they "soften," they can be more dangerous than a high GH. Cited verbatim from Dr. Neale Monks:
“Domestic water softeners do not produce soft water in the sense that aquarists mean. What domestic water softeners do is remove the temporary hardness (such as carbonates) that potentially furs up pipes and heaters by replacing it with permanent hardness (such as chlorides) that does not. While you can pass this softened water through a reverse-osmosis filter to remove the permanent hardness as well, until you have done so, you shouldn't consider the softened water as being suitable for soft water fish.
In fact, aquarists are divided on whether the resulting softened water is safe for keeping fish at all. The odd balance of minerals in softened water is not typical of any of the environments from which tropical fish are collected. While the chloride levels are much higher than those soft water fish are adapted to, the levels of carbonate hardness are too low for the health of hard water fishes like Rift Valley cichlids, goldfish, and livebearers. So the safe approach is not to use it in any aquarium, and instead draw water from the unsoftened drinking water source in the kitchen.”
The above is largely taken from an article I wrote some years ago for another site, but it is now on AbbeysDad's blog, it may give you more detail.
