Most people measure their tap and tank water using two test kits, GH and KH.
GH stands for General Hardness (Total Hardness) and is a measurement of total multivalent cations in the water.
KH stands for Carbonate hardness (Temporary Hardness) and is a measurement of multivalent cations that are carbonates and bicarbonates.
Both GH and KH are expressed in equivalents of CaCO3
(A multivalent cation is a positively charged ion with two or more positive charges, ie, Ca++, Mg++, Al+++, Fe+++)

What most people don’t realise is that the aquarium KH test kits don’t actually measure KH, but Alkalinity instead. Alkalinity has nothing to do with water hardness but is the buffering capacity of the water defined as the capability to neutralize H+ ions. This Alkalinity can come from many different compounds, but carbonate alkalinity tends to make up most of the alkalinity in our tanks. Other compounds that can contribute to alkalinity are phosphate, nitrate and silicates just to name a few.

Coming back to water hardness. The following 3 types of hardness are the normal way of expressing water hardness.
Total hardness (GH) is the total amount of multivalent cations in water and can be split into two parts, temporary and permanent.
Temporary Hardness (KH) is the amount of multivalent cations that are carbonates and bicarbonates.
Permanent Hardness is the amount of multivalent cations that are everything except carbonates and bicarbonates, ie. chlorides, sulphates, etc.

Total Hardness = Temporary Hardness + Permanent Hardness

Temporary Hardness (KH):
To lower temporary hardness is fairly easily done. Boiling water will effectively remove it as the insoluble carbonates formed from the thermal breakdown of bicarbonates can be filtered off. To raise KH you can add calcium or magnesium bicarbonate. Adding baking soda (NaHCO3) will not raise KH as sodium isn't a multivalent cation, but will raise alkalinity as described above.

Permanent Hardness:
Lowering permanent hardness can either be done by diluting it with softer water, ie. rain water and RO water or by filtering it through a deioniser or RO system. Be aware that many domestic water softening ion exchangers replace the multivalent cations with sodium ions. This type of water softener should not be used for planted tanks. Raising permanent hardness can be done by adding calcium or magnesium sulphate.

Total hardness (GH):
As GH is a combination of both temporary and permanent hardness, making any changes to these will affect the GH. So either boiling water or using a deioniser will lower GH. Adding any calcium or magnesium salt will raise GH.

So what does all this mean for our planted tanks?

GH or Total Hardness test kits are generally very good and do measure what they say they measure. GH is a good thing to have as plants need both calcium and magnesium present to thrive. If your water has a very low GH then this can be increased by adding GH booster or similar products, which normally contain calcium and magnesium sulphates. Most plants do fine in a wide range of GH’s so unless you have exceptionally hard water don’t worry about it.

It’s KH test kits that are the problem as they are not measuring water hardness at all but alkalinity. This doesn’t really matter as we aren’t interested in KH anyway as alkalinity is much more important to us. Plants don’t require alkalinity to grow so you could keep your tank at zero, but this requires a lot of water changes to keep tank conditions stable as other alkalinity affecting compounds may build up. Some plants and fish don’t like very high levels of alkalinity so if you wish to keep these then the best way to lower it is by blending with reverse osmosis water. Fish are very sensitive to changes in alkalinity so any changes must be done slowly over a period of time.

It is possible to lower the alkalinity by adding a strong acid such as hydrochloric acid as this adds H+ ions to the water. I have heard of some people adding strong acids to their tanks to lower the alkalinity but I highly recommend you don't try this if you have any livestock.
Adding CO2 to the tank doesn't affect alkalinity (or KH test kits) at all. This is because equal amounts of H+ and HCO3- are formed so in affect cancelling themselves out.

Wikipedia - Hard Water

Wikipedia - Alkalinity