I want to hide page extensions like stackoverflow does. How does the following work?
http://stackoverflow.com/tags/foo
http://stackoverflow.com/tags/bar
There are a couple of ways to do it under Apache+PHP, but the essential principle is to make a set of URIs (perhaps all URIs, depending on your site, but you may want different scripts to handle different portions of the site) translate to a single PHP file, which is told what object the user has requested.
The conceptually simplest way is to rewrite every URL to a script, which gets the URI through $_SERVER['REQUEST_URI'] and interprets it as it likes.
The URI rewriting can be done with various methods including mod_rewrite, mod_alias and ErrorDocument (see Apache docs).
Another way is to set up more complex URL rewriting (probably using mod_rewrite) to add the path as a GET variable.
There is also the $_SERVER['PATH_INFO'] variable which is loaded with the non-existent portion of the path. This option requires little or no modification to Apache config files, but reduces the flexibility of your URLs a little.