Websafe colour palette
Note: Because computer monitor resolutions have improved (and even mobile devices have good colour palettes now), it's no longer necessary to use web safe colours in your website design. There's still quite a bit of interest in them, though, judging by the number of people who visit this page, so I've left this archive article otherwise intact.
To be confident that your colours will come out as intended across different platforms and browsers, you need to pick them from the websafe colour palette. Here it is, with the colour numbers in the middle of each box: