In layman's terms, what is reverse osmosis filtration?
A RO filter filters water by pushing water it through a semi-permeable membrane (think of it as a piece of plastic that can let water (but not much else) slowly ooze through). 

Normally, if you separate dirty water and clean water with such a membrane, natural forces will cause the clean water to slowly move over to the dirty side. This  process is called Osmosis. (this is an oversimplification, not all impurities cause osmotic pressure).

In a Reverse Osmosis filter, a pump is used to pressurize the water on the dirty side, which forces the water to go the opposite direction...from the dirty side to the clean side.  i.e. it's Reversing the Osmosis. This clean water is then collected and that's what you drink.

Eventually the water on the dirty side becomes so full of impurities that it's hard to push any more water through the membrane, so that water is drained away and replaced with fresh water. This is why RO filters use so much water -- they waste 4 to 20 gallons of water for each gallon of purified water they produce.

Most RO filters also have a conventional pre-filter (or multiple filters) to remove a lot of the contaminants before they reach the RO membrane so the RO membrane is only removing the dissolved impurities that are hard to remove with a filter.