When lighting designers talk about this they are referring to the spectrum of the light, and their control over it. More expensive fluorescent lamps use 3 or more phosphors so that the light they produce not only appears to be the color they want it to be, but also the stuff illuminated by the lamps looks realistic (or at least "pretty"). Smart retailers have used "tri-phosphor" fluorescent lamps for a long time now to make their products look as appealing as possible. Nice new LED (expensive) lamps do the same sort of thing except with multiple colors of LEDs.
Low cost "white" LEDs are blue LEDs with additional yellow phosphor doping. This makes the resulting light appear bluish-white. The stuff the light is shining on, however, may not appear natural, particularly if it is colored red or orange, among others. These colors reflect far less when illuminated by these lamps than one would expect, and don't look right. This may not matter in a utility lighting situation, but is not desirable as a replacement for the nice broad spectrum of incandescent lamps. Using more colors of LEDs or more types of phosphors can give better "rendering" to illuminated objects and people.
Here's an example. I have one of the pricey LED lamps. These use 4 different colors of LEDs in a large emitter array. They're available in several different whites, so you can pick one you like. I put these in a track fixture in the kitchen, replacing a PAR CFL lamp. Suddenly, the carrots appeared a much more vivid orange. I can test this, by moving the carrot back under a remaining fluorescent or incandescent lamp, and the difference is striking. I would replace all 5 lamps in that fixture with those LED floods, but at $90 a piece, that may have to wait a while.
You can do your own experiments on color quality, etc. with an optical spectrum analyzer, aka "a prism". With sunlight, you'll see the familiar rainbow. Incandescent lamps will produce a similar rainbow, but with stronger oranges and weaker blues. Now try it on some LEDs and fluorescent lamps, and you can see distinct color lines. Three or 4 lines will render more realistically than 2. Congratulations, you are now a quantum physicist!;)