Distillation is the brute-force way to remove impurities from water.  A distiller boils water, then cools the steam until it condenses.  The resulting distillate drips into a jug.  Salts, sediments, metal ions, anything that doesn’t boil off or evaporate, stays behind in the boiling pot.  A broad group of chemicals known as “volatile organics”, which contaminate some groundwater, include chloroform and benzene which pass through distillers and end up in the product water.

Distillers are designed to work on a countertop.  Most stills do an excellent job removing high concentrations of minerals, from ones as harmless as calcium to those as toxic as lead.

Time and Money
Distillation is a slow process.  110 volt stills require about 5 hours to make a gallon of distilled water.  Also, considerable energy is expended converting a gallon of water into steam.  At the current national average electricity rate of 7.75 cents per kilowatt-hour, distillers use about 24 cents worth of electricity to distill a gallon of water.

You might welcome the heat from a distiller in the wintertime.  In the summer, though, the heat will make your air conditioner work longer -  each gallon of water distilled will add 6 cents to the cost of running an air-conditioner.

Recommendations
A distiller makes the most sense if your water supply is brackish or polluted only with heavy metals.  A distiller removes salt very well, and it’s better than a reverse osmosis system for removing heavy metals.  A distiller won't eliminate organics such as pesticides - use a carbon filter or a reverse-osmosis system instead.  To remove high levels of calcium and magnesium, use a water softener, not a still.