Distilled water is water that has been boiled and turned into steam. That steam then has been cooled down and condensed back into water leaving some of the contaminates behind. What it will do is kill any bacteria and any viruses because of the temperature need to boil the water, along with removing any metals as mercury and lead.
There are many chemical contaminates that will turn to steam and follow the water steam to the condensed (clean) side. If you have city treated water the chlorine will pass through along with any herbicides and pesticides that may be in the water.
Distilled water will also absorb carbon dioxide from the air. This will make the water acidic. High acidic body is not a good thing. This can cause low energy to acceleration of free radical damage to your cells, and many other things to go wrong in-between. But if you eat a good balanced diet, this is nothing to worry about. Also keep your distilled water in a full sealed container. It can not absorb what it does not touch.
Distilled water has no minerals in it. This makes it very good to use in anything that heats up water like a coffee maker or iron. No left over calcium deposits to plug up the works. There are a lot of controversies over if this is bad for your health. But the minerals that your body needs do not come from your water intake. It is from your foods that you eat.
Sodium Fluoride is removed from distilled water. This is the stuff that is supposed to keep your teeth from getting cavities. But many studies have suggested that it may do that at a very high cost to your health. This may be the biggest reason to distill your water.
I have been drinking distilled water now for 5 years. I will only if nothing else is available drink tap water. I do not make it a habit of eating out very much, so I get little restaurant water. After distilling my water I always run it through a Pur water pitcher. This makes it taste better and helps in removing any chemical pollutants left behind used by industrial and agricultural polluters.