Color (syntax highlighting) within an HTML <code> tag
It's a Javascript library. There are quite a few out there, the most popular being SyntaxHighlighter. My personal favorite is Chili, though.
There is an excellent FAQ What is syntax highlighting and how does it work? over on meta.SE.
It’s pretty extensive. For your convenience, i’ll quote the part most related to your question:
How does it work?
When creating or editing posts, syntax highlighting is assigned to the preview based on question's tags as soon as you stop typing for 5 seconds.
Stack Exchange does not have its own syntax highlighting engine. It uses highlight.js, and may not necessarily be using the latest release of that library. Therefore, any bugs and feature requests regarding syntax highlighting cannot be handled by Stack Exchange.
Why isn't my code being highlighted correctly?
[ Continue reading the original post … ]
Stack Overflow uses Google's prettify JS library for doing syntax highlighting. It executes on the client side after the HTML has been delivered by the server. That's why you don't see it in the raw HTML source. If you have a browser plugin such as FireBug, you'll be able to inspect the DOM after prettify has done its magic.
Update 2020-09-14: Stack Overflow switched from Google's prettify to highlight.js.