How Do LEDs Work?

I've watched lightbulbs change a lot in the past decades, and with each technology change I've had to re-learn how they work. Incandescent bulbs were everywhere when I was little, and some had already been burning for decades. I understood how those worked; we played around with filaments in school. Then along came compact fluorescent (CFL) bulbs, and I learned how those worked. Now, I'm slowly installing LED-powered bulbs. I'm surrounded by LEDs, both as indicator lights and in my TV, computer monitor, phone, and pretty much every other gadget. But how do they work? And why do some LED lightbulbs have huge heat-sinks on them?

This two-minute video from Physics Made Fun gives a very nice overview, breezing through the subject and even explaining why LEDs have different colors. Behold:

If you want a much longer, slower presentation, check out this video from the Edison Technology Center. Around 3:30, this video shows how OLED backlights are built.