Color Temperature and CRI (an abbreviation for Color Rending Index, typically written as "CRI" with no dots between the letters) are terms you will hear when someone is referring to actual visible color of a light source, and/or how the light source or luminaire makes other colors look when they are illuminated in a room space. The main difference in the two terms is Color Temperature is a measure of how the actual light source appears to your eye, and CRI is a measure of how the various colors in the room space are perceived when illuminated by that light source.
Color Temperature is usually a reference to how a white light source appears to the eye, and is expressed using a numerical figure in terms of degrees KELVIN (or "K"). Sometimes a user may hear Color Temperature expressed in its more technical terminology, Correlated Color Temperature (CCT). A light source that looks to the human eye like it is more yellow/reddish in color, or more "warm" in tone like a sunset will have a lower color temperature such as 2000K. White light sources that are more blue or "cold" in color tone like the color of fluorescent lighting in an office building would have a higher color temperature such as 5000K.
Unlike some older bulb technologies such as incandescent and fluorescent, LED technology is extremely flexible and able to be designed to achieve almost any Color Temperature.
Since most USA homes have been accustomed to being illuminated by INCANDESCENT bulbs, which were themselves originally designed intended to mimic the color of candlelight; the warmer colors in the range of 2200K to 2700K are preferred by many USA consumers to create a cozy warm feel in their home environment. In order to help consumers understand these numbers, colors in this 2200K to 2700K range may be called "Warm White." When the color temperature numbers approach 3000K, which is also a very popular and common color for LED bulbs and downlights, the color becomes more similar to what consumers are accustomed to seeing with halogen bulb technology. The color of the white light source at 3000K is more of a true white appearance and has less reddish/yellow components. However this 3000K does not yet veer into the more "blue-ish" tones. Some consumers prefer this 3000K color over warmer 2200K to 2700K colors, and like to light of such color in downlights, undercabinets, table lamps, and any other home or hospitality lighting.
At 3500K to 4000K, the white light starts to look more like a fluorescent light source, and these colors are not typically preferred in most residential environments except laundry, garage, and work spaces. 3500K to 4000K bulbs are more suited to work or commercial spaces than most home environments. Colors above 4000K are not typically used anywhere inside a home, but may be used in outdoor fixtures. 5000K is often used in products such as streetlights and roadway.
The most common color temperatures for residential LED bulbs are 2700K and 3000K. Both color temperatures are quite popular for USA residential use, and preference of one over the other varies greatly by user. Although both of these colors may be described as "warm white," 2700K is the warmer of the two options and will appear a bit more yellow/reddish in comparison. 2700K looks more like traditional incandescent bulbs, and 3000K looks more like traditional halogen bulbs.
Color Rendering Index, or CRI, is a measure of how the colors in a space are perceived when illuminated by a light source. CRI is represented by a number from 0 to 100 that identifies the ability of a light source to show colors in the room space "naturally" when compared to a reference light source such as natural sunlight or incandescent. Objects or colors illuminated by light sources with lower CRI values may appear less vibrant that those that are illuminated with higher CRI values. Early fluorescent products were notoriously panned by many consumers because they caused one's appearance to look "unnatural" under their fluorescent glow. Much of the reason behind such an "unnatural" appearance under fluorescent lighting was due to the low CRI values of early fluorescent bulb technology. Incandescent lighting is 100 CRI, and many of the early fluorescent products were less than 60 CRI, meaning most colors wouldn't look comparatively very vibrant under these low CRI lights. Particularly red colors ,which are important in allowing one's face to look natural under illumination, require light sources that have higher CRI values to look optimal when illuminated.
Most LED bulbs sold today are typically at least 80 CRI, and lights with such CRI values have been found to be generally acceptable to most consumers. Some users may want a higher quality of light that renders colors more closely like that of an incandescent bulb. Light sources with at least 90 CRI are typically called "High CRI" light sources, and such light sources render colors very well and create environments similar to lighting using incandescent. However, "High CRI" bulbs typically have a higher price point due to the added technology required to create the higher CRI light quality. Some home spaces such as kitchens and bathrooms will benefit from utilizing light sources with higher CRI values such as 90 CRI since meat, fruit, skin tones, etc. will all look more vibrant under "High CRI" light sources. The state of California has mandated specific light quality requirements including a CRI minimum of 90CRI for most LED light bulbs used in all new residential construction.