You can view the source of the examples to see how they are done.

Ajax Username Validation with Prototype
Ajax Username Validation with jQuery

note: more examples will come in time. I am currently reading the mootools, dojo, and mochikit.

The only PHP code you will need is as follows:

include("../db.php");
$name = ereg_replace("[^A-Za-z0-9-]", "",mysql_real_escape_string(strip_tags(trim($_GET['name']))));
if(empty($name)) die();

$sql = "SELECT `username` FROM `common_usernames` WHERE `username` = '$name'";
$query = mysql_query($sql,$db);

if(!mysql_num_rows($query))
{
	die(''.$name.' is available!');
}
else
{
	die(''.$name.' is taken!');
}