Water softening

Water softening is an aspect of chemical filtration, for it involves reducing certain dissolved minerals in the water, though not all dissolved minerals contribute to hardness. Reducing the carbonates and phosphates will reduce the alkalinity or "buffering" that keeps the pH stable. Reducing the ions will lower the water's electrical conductivity. Diluting the total dissolved solids with distilled or reverse osmosis ("R/O") water is one way to achieve these objectives. Rainwater is a cheaper, low-tech alternative. Peat filtration, if the water is lightly buffered and only a few points of softening are expected, is also something to look into. Sometimes your water board has softened municipal water right at the source, often with the "lime/soda" technology, which result in soft water with a temporary high pH.
Reverse osmosis: a green thought. Gallon for gallon, reverse osmosis technology, once you've installed it, creates several gallons of wastewater for each gallon of desirable filtrate. What will our great-grandchildren say, when they hear that, in order to obtain a gallon of water stripped of its mineral solutes, we were flushing into the sewer five or six gallons of filtered, pre-treated, chlorinated drinking water— rather than trap and filter rainwater? That we saw nothing in this technology to make us uneasy? (When they hear that we were even flushing our toilets with purified drinking water, they'll figure we were pretty benighted anyway.)
Partial softening by boiling. Some low-tech water-softening can be achieved on a small scale simply by boiling. (Ingenuity and exertion so often can substitute for cash.) If you are wary of rainfall and currently saving up towards your costly reverse-osmosis installation, you can still have partially softened water, as long as all you need is a few gallons. Just boil tapwater in a large partially covered aluminum, stainless steel or enamel pot, the one you use for spaghetti or for a crab boil. As the water heats up, the carbonic acid (H2CO3) formed by dissolved carbon dioxide is reduced as the CO2 is driven off.  Chemical balance is maintained as dissolved bicarbonate breaks down into CO2 and carbonates (e.g. CaCO3). Carbonate isn't soluble like bicarbonate, so it precipitates out: it's the white deposit that coats your kettle. (In a similar reaction, Ca and Mg carbonates in hard water can precipitate as a gritty white coating on your submerged aquarium heater.)
Let the boiled water settle out its precipitates for a couple of hours, then siphon it off into your storage container. If you were to leave the water in contact with the carbonate deposits long enough, atmospheric carbon dioxide would slowly diffuse into the depleted water, where it would dissociate into carbonic acid and begin to reverse the process, to re-establish the former equilibrium. That's the reason for drawing it off  while it's still quite warm 
The hardness that can be precipitated by boiling is often called the "bicarbonate" or "temporary" hardness. Boiling will not eliminate the other components of the GH (General Hardness), such as calcium and magnesium chlorides and sulfates, nor will it affect nitrates, phosphates, silicates or heavy metals. As it is, this boiled water will be utterly depleted of its component of dissolved oxygen, which boiling will have driven off, so once it's cool you'll need to run an airhose in it for a day or so, to diffuse oxygen into it again, before it's ready to mix with aquarium water.
Here's a good tale that's apropos: A fishkeeper with extremely alkaline well water (240 ppm KH) found that when his malfunctioning heater sent tank  temperatures soaring over 85°F, calcium carbonate precipitated onto plants and every inanimate object in his tank. After the disaster, KH was found to have dropped to 180 ppm! My hunch was that apparently the heat was enough to drive off sufficient carbon dioxide, so that the plentiful dissolved bicarbonates broke down into CO2 and insoluble carbonates. This misfortune could only be an issue at extreme KH levels, but it suggested to me that he could routinely achieve some softening without actually boiling his extremely alkaline water, just with a heater in a 55 gallon container. I wish one of you with sufficiently hard water would do some simple experimenting and e-mail me about the results you get.
Peat filtration. Alternately, soft water combined with high pH is a situation where I think peat filtration would work to drop the pH, all the more effectively because the buffering is already so low: Peat filtration has its own page. Pre-wetted sphagnum peat also softens water slightly, and the way it does this has the effect of reducing the pH. The complicated organic molecules of humic substances in peat, one of my substitutes for handy but expensive "blackwater extracts",  bind the positively-charged calcium and magnesium ions and exchange them for positively-charged hydrogen ions. The more free H+ ions, the lower the pH, by definition.  Other sources for mild "blackwater" chemistry with some similar effects are leaf litter, Osmunda fiber if you can get it, coconut shells, even a used green tea bag in the filter.
In general, keeping in mind that stability of the pH is more important to fish than any particular pH value, you won't rush too fast to add more peat: use a little, then test it and wait a day. You should avoid making changes greater than 0.2pH in a day. Keep a log.
Ion-exchange water-softening resins. In common household system water-softening systems, ion-exchange resins swap two sodium or potassium ions for each divalent calcium or magnesium ion they take up. The resin beads are recharged in brine.

As a general rule, if the ion-exchange resin is "re-chargable" in salt brine, then it's putting sodium ions into your water. Good for lathering your shampoo, but not for aquarium work. Even if it is for sale at the LFS. Some aquarists dismiss the effects of this added sodium. Others tell you that household water softened by ion-exchange is never suited to aquariums. Sodium is undesirable in a freshwater aquarium; but in a planted tank there is a better cation than sodium for recharging ion-exchange resins: potassium.
Recharging with potassium chloride. Though it is more expensive than common rock salt, potassium chloride (KCl) is just as effective at recharging ion-exchange resins, according to a Cargill Salt website FAQ concerning water softening. The Q&A in question goes like this:
"What is the difference between sodium (chloride) and potassium (chloride)?"
"Both do the same job. They replace calcium and magnesium on the softener resin during the regeneration process. When you use sodium chloride, sodium will be added to the soft water during use and when you use potassium chloride, potassium will be added to the soft water. People whose physicians have advised them to eliminate sources of sodium from their drinking water normally use potassium chloride. In some people who have kidney or other renal problems, potassium can aggravate those problems. Most healthy people (>97%) can use sodium chloride without trouble and sodium chloride is less expensive. If you have any questions, consult your physician."
Another good archived article encouraging you to substitute potassium for sodium is "An Alternative to softening with sodium", from Pipeline. a newsletter of the National Small Flows Clearinghouse, Winter 2001.
When substituted for rock salt, potassium chloride works in the same way to replace Ca and Mg ions. In a planted tank potassium is more desirable than sodium: the additional potassium would be used by plants, so that there should be no build-up to be diluted by water changes. In fact, dosing with potassium, often scarce in fresh water, is central to the "Sears-Conlin" method of controlling algae by limiting phosphate in planted tanks.
You can get potassium chloride in the form of salt substitutes such as "Nu-Salt."  Though I don't need to soften water here in New York, I don't understand why recharging ion-exchange resins with potassium chloride isn't perfectly standard aquarium practice.
Other ion-exchange resins suitable for chemical filtration in aquariums work in a way similar to peat, by adsorbing calcium and magnesium ions, thus softening the water, and by releasing hydrogen ions in exchange. The additional hydrogen ions increase the water's acidity, and the pH value is consequently lower. This type of ion exchange does not increase levels of sodium or potassium in the water.
Biological water-softening processes. Plants take up calcium and magnesium in building and maintaining their structures, and they store them, thus functioning ecologically as a reservoir for these elements, like a standing forest that locks up carbon for a time. When plant stems and leaves are pruned or siphoned out, they are "exported" from your aquarium ecosystem. On the other hand, if they soften and disintegrate within the aquarium, their molecular building blocks remain part of the system and get recycled.
Snail shells similarly store calcium, but this gets returned to the system if the snail dies and you don't remove the shell. But when you flush Melanoides snails out of your filter medium and discard them, you are also "exporting" from your closed system the calcium represented in their shells. Of course, on a practical level, this isn't really a terrific way to achieve any notable water softening. So you may snicker. But the principle of "exporting" from the system is important to keep in mind, because in the end, "exporting" the ions like Ca and Mg that make water "hard" is the only way of permanently softening water.