What are the differences between Single Stage, Dual Stage and Reverse Osmosis Under Sink Water Filters?
Single versus dual stage is easy; dual stage is essentially two single-stage filters in succession. Bulkier, more expensive. Better removal of contaminants. Installation is the same. 

Reverse osmosis is an entirely different technology. It generally has a prefilter because RO doesn't deal with particulates, only dissolved chemicals. It does handle those very well. It works by using pressure and a semipermiable membrane to achieve the purification. The membrane lasts a long time, but the system uses additional water to concentrate the impurities and flush them away so there's some ongoing cost there. Installation is a bit more complicated as well.