I've got the following code to work now, but there are still a few problems:

	$("#action_button").click(function() {
		$('#success').append("Hello");
		var username = $("#username").val();
		var password = $("#password").val();
		var dataString = '&username=' + username + '&password=' + password;
		if(username=='' || password=='') {
			$('#success').fadeOut(200).hide();
			$('#error').fadeOut(200).show();
		} else {
			$.ajax({
			type: "POST",
			url: "processing/logsig.php",
			data: dataString,
			json: {session_state: true},
			success: function(data){
			if(data.session_state == true) {
				$('#success').append("True");
			} else if(data.session_state == fale) {
				$('#success').append("False");
		   	}
		  }
	   });
	  }
	});

It'll process the Php script, but it won't process the JSON... I've got 		$('#success').append("Hello"); working, but I can't seem to get the ajax json data return to work... It processes the php script because when I reload the page after clicking the action_button, it loads the page that is supposed to come after the user logs in. Any ideas? Also, below is the revised php script:

    <?php
    header('Content-type:application/json');
    session_start();
    include("global-settings.php");
    mysql_connect($dbhost, $dbuser, $dbpass)or die("Could Not Connect: " . mysql_error());
    mysql_select_db($dbname) or die(mysql_error());

    $email = mysql_real_escape_string(strip_tags($_POST["username"]));
    $password = sha1($_POST["password"]);
    $sql = "SELECT * FROM users WHERE username = '{$email}' AND password = '{$password}'";
    $result = mysql_query($sql); // or exit("ERROR: " . mysql_error() . "<br>IN QUERY: " . $sql);

    if (mysql_num_rows($result) > 0) {
    	$row = mysql_fetch_array($result);
    	$_SESSION["userid"] = $row['user_pid'];
    	$json = json_encode(array('session_state' => true));
    	echo $json;
    } else {
    	$userid_generator = uniqid(rand(), false);
    	mysql_query("INSERT INTO users (user_pid, email, password, datetime_registered, is_leader) VALUES ('$userid_generator', '{$email}', '{$password}', NOW(), 'no')");
    	$id = mysql_insert_id();
    		$leaders = mysql_query("SELECT * FROM users WHERE is_leader LIKE '%yes%'");
    		while($rows = mysql_fetch_array($leaders)) {
    			if ($rows['is_leader'] == 'yes') {
    				$leader_id = $rows['user_pid'];
    				mysql_query("INSERT IGNORE INTO friends (node1id, node2id, friends_since, friend_type)
    				VALUES('$leader_id', '$userid_generator', NOW(), 'full')");
    				}
    	$_SESSION["userid"] = $userid_generator;
    		}
    	if(is_dir($userid_generator)) {
    		echo "Something wen't wrong. A bug report has been sent and we are doing what we can to fix it.";
    		$message = 'Registration problem on account number $userid_generator. The user succesfully registered, but there is already
    		a directory with the account id of $userid_generator.';
    		mail($bug_report_email, "Registration Bug!", $message);
    	} else {
    		mkdir('media/User-PID{' . $userid_generator . '}', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/photos', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/backups', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/videos', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/documents', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/developer', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/developer/apps', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/developer/themes', 0777);
    		mkdir('media/User-PID{' . $userid_generator . '}/xml', 0777);
    	}
    	$json = json_encode(array('session_state' => false));
    	echo $json;
    	}
    ?>