The first white LEDs weren't explicitly white LEDs. Instead, red, green, and blue LEDs were packed into a single LED enclosure. However, if you mix red, green, and blue light, you will get white light, it's really just a matter of getting the correct proportions of different colored photons.
This remains the standard for RGB LEDs, and some have even experimented with improving the range of color these LEDs can produce. The human eye is extremely sensitive to green frequencies of light, and by adding a fourth LED to a package — it's best called 'emerald', or a slightly bluer shade of green than what we're used to in green LEDs — you can make an LED with a wider color range, or if you prefer, a whiter white.
This was the first method of developing a white LED, and while the LED light bulbs you pick up at the hardware store don't have individual red, green, and blue LEDs inside, it is still a fantastically popular way of creating more colors with LEDs. Those neopixels, WS2812s, or APA101s, all have red, green, and blue LEDs tucked inside one enclosure. Some of the more advanced individually addressable RGB LEDs even add a fourth LED, for white. But how are those individual white LEDs made?
The first white LEDs, made without three individual LEDs, were made with the magic of phosphors. Phosphors are a well-understood science, most commonly found in lighting applications in fluorescent bulbs. Fluorescent bulbs don't produce white light on their own, they produce ultraviolet light by exciting mercury vapor. However, by coating the inside of a fluorescent bulb with a powder, this ultraviolet light can be converted into red, green, and blue light. The result is a fluorescent bulb that lights your garage or workshop.
This can be done with LEDs as well. With an ultraviolet or violet LED packaged inside a phosphor-coated enclosure, you can make a white LED. This is known as a full-conversion white LED.
Full conversion LEDs are inefficient, though, so by the mid-90s the race was on to create a partial conversion LED. This type of LED would illuminate a phosphor with blue light, and the phosphor would convert a portion of that blue light into something broadly yellowish that contains a mixture of red and green wavelengths. Adding more red phosphors to the mix creates “warm white” LEDs.
In 1996, the Nichia Company announced the production of white LEDs, and the rest is history. For the past twenty years, the power, efficiency, and brightness of these LEDs has increased. Now, or in the very near future, the default for home lighting won't be incandescent bulbs, but powerful LEDs consuming a mere fraction of the energy an old Edison bulb would, shifting more of the task of keeping your house warm onto the heater.