Here's the juice as far as I know:
A) Distillation. This is the process by which water is boiled and the steam is recaptured. The assumption is that water has a lower boiling point than most of the impurities contained within it and as a result, those impurities are left behind in the boiling chamber. This results is fairly pure H2O. There are, however, atoms and molecules that escape the process or that are added back in through the cooling tubing (usually Copper = bad for inverts) used to recapture the steam.
B) Reverse Osmosis. This is the process by which water is forced through a semi-permiable membrane which removes certain molecules from the water. This is a good methodology as well, but as with distilled water, there are molecules that escape the process.
C) De-Ionization. Removes an-ions and cat-ions from the water that neither Distillation nor RO alone can achieve. This results in the purest H2O that can be achieved on a practical level, but by itself it's expensive in both water usage (one gets about 1 gallon of DI water per 3 gallons consumed) and filter membranes.
The best way to get the purest water as I understand it, is to use a combination of both RO and DI. There are two benefits to this methodology:
1) RO removes most of the impurities quickly and fairly inexpensively.
2) DI removes, for all practical purposes, the rest. ROing the water first speeds the process up and lowers the cost in both water and membrane consumption on the DI end of it.
Depending on your initial source, there are many ways to filter your water. I use a 5 stage process which involves the following in chronological order:
1) Sediment filter (takes out crap you can see)
2) Activated Carbon
3) Reverse Osmosis
4) De-Ionization
5) Dedicated Silicon remover
