Quickly...
Water, under pressure, passes through a 1 or 5 micron prefilter then through a carbon block filter, up through a Reverse osmosis filter then through the Dionization filter then out to your container. There is also a 'waste water' line coming off of the RO Filter which you can either put down the drain or collect for other uses (watering plants, washing cars, etc.)
That would be a crash course. Most of the scientific stuff happens in the RO and DI chambers. That would require more than a crash course.