How do I run PHP code when a user clicks on a link?

Yeah, you'd need to have a javascript function triggered by an onclick that does an AJAX load of a page and then returns false, that way they won't be redirected in the browser. You could use the following in jQuery, if that's acceptable for your project:

<script type="text/javascript" src="jquery.min.js"></script>
<script type="text/javascript">
function doSomething() {
    $.get("somepage.php");
    return false;
}
</script>

<a href="#" onclick="doSomething();">Click Me!</a>

You could also do a post-back if you need to use form values (use the $.post() method).

As others have suggested, use JavaScript to make an AJAX call.

<a href="#" onclick="myJsFunction()">whatever</a>

<script>
function myJsFunction() {
     // use ajax to make a call to your PHP script
     // for more examples, using Jquery. see the link below
     return false; // this is so the browser doesn't follow the link
}

http://docs.jquery.com/Ajax/jQuery.ajax