It's called a TDS meter. It records the total dissolved solids in the water by using tiny metal electrodes that react with the ions. The reading gives the measurement in parts per million.
The zerowater people say that you should change the filter when you get a reading of 006 ppm. However, I change mine as soon as it reads 001 because the typical fluoride concentration in tap water is 1 ppm. At least that's what I've seen around the country.
I don't even really need the meter so much anymore. I notice the chlorine taste immediately when it's time to change.
I tried using a distiller years ago, but I never could get everything out of the water. I believe some of the chemicals used to treat water have similar evaporation/condensation points so that when the water evaporates and re-condenses in the collection jug, the chemicals do too. It still tasted of chlorine.
I've also tried buying store-brand distilled water, but it was even worse tasting. It tasted like a chemical soup.