Back in the days of 8-bit computers, color palettes were necessary because memory was scarce. Nowadays any computer can display true color, or close enough, and we have all the storage we could possibly want for storing big assets, but pixel artists still use color palettes, especially for games. The reason is that a limited, but well-chosen color palette can set the mood of the artwork and make it feel unified. But outside of pixel art communities it can be hard to find much information on the topic.
Dawnbringer’s 16-color palette: possibly the best known and most appreciated by pixel artists these days.
Dawnbringer’s 32-color palette: successor to the above, not quite as famous.
The Tango Icon color palette: designed for GUIs, it’s nevertheless just as usable for games or websites.
An intriguing one is Ametameric, a version of the 16-color ANSI palette for people suffering from color blindness.
This thread about color palettes on Open Game Art lists a bunch more; over on the Lemma Soft forums, someone converted the entire SVG palette to Ren’Py format, and fluffy points me at this list.
Last but not least, Wikipedia has several lists of color palettes, such as this one. The MacIntosh II and RISC OS palettes are especially beautiful. And lospec.com has an entire palette database.