Just to make a little correction to one of the above answers. Chlorine destroys Reverse Osmosis membranes. Therefore you'll always see a carbon filter that removes chlorine (or chloramine, depending on your local water supply) before it goes through the membrane.
A further filtering step is often the addition of mixed-bed anion/cation resins after the RO step. This is known as a deionization system (or DI for short). This produces water with virtually no ions in it. Not totally safe for drinking for esoteric reasons of increased reactivity, but as low in dissolved solids (pure) as can be easily obtained (outside of distillation, or other methods.)
You can easily find inexpensive dissolved solids meters on the web. These can give you a numeric reading of dissolved solids, but won't tell you what they are. They can read a very high number for hard water, and a low number for soft water, lower still for RO water, and essentially 0 for DI water.
As has been noted above, all these filtration systems are only good for low flow situations, unless you have a storage tank (then without something like a UV light, bacterial contamination becomes an issue.)
Alan Lichtenstein, MD
Mens et manus
He who laughs last...Thinks slowest.