The principle of LEDs is basically the same as that of any other diode-- current, in the form of electrons, flows from one side to the other, but not back. OK, technically the electrons go the other way around, because they're negative, and it's really confusing but its codified in convention and there's nothing we can do about it*. Anyway, the particular materials used in LEDs can also be charged by current with a large number of "holes." These are not physical holes, but rather a sort of "anti-" electron**. They're empty spaces in atoms that normally would hold electrons, but for whatever reason don't. When the electrons meet the holes, they "fall in." This "fall" is a lot like a physical fall, in that it releases some potential energy, in the form of a photon***. The amount of energy released determines the energy of the photon, and therefore the wavelength, and therefore the color. Some colors (red) are a lot easier to generate, whereas some (blue) require some fairly complicated chemicals and materials to create. White LEDs often rely on a two-stage approach, generating blue and ultraviolet photons that also activate a phosphor coating that in turn generates yellow photons, which mix with the blue to get something whitish.
*Damn you Benjamin Franklin!
**The actual antielectron is a positron, which is completely different, being a particle with the same properties as an electron except for opposite charge. Holes are just absences of electrons.
***I accidentally typed "phorm of a photon" here, but decided to remove it. You're welcome.
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