Put simply, quantum dots are a new technology that allow today's TVs to create much more vivid and saturated colors than ever before.
Currently found mostly in high-end offerings from Samsung, LG, and Sony, quantum dots (also sometimes referred to as "nano-crystals") are microscopic dots that react to light by emitting color. Manufacturers print a sheet of dots onto a TV display panels, which allows them to produce unusually intense colors—especially reds, greens, and cyans.
Quantum dot technology represents a major improvement for TV color, and sets equipped with it can produce richer pictures than any past models were capable of. On the other hand, because it's a completely new technology in TV world, it also ratchets up prices. Over time, quantum dot will trickle down to entry-level models, but for now you're only likely to find it on flagships.