Do not use mysql_* functions. Use mysqli_* instead. When should I use MySQLi instead of MySQL?
Result page:
<table class="users">
    <tr>
        <th>NAME</th>
        </th>Register Time </th>
    </tr>
<?php
$query = mysql_query("SELECT * FROM `register`");
$lastId = 0;
while ($row = mysql_fetch_array($query)) {
    $lastId = $row['id'];
    echo '<tr>';
    echo '<td>'.$row['name'].'</td>';
    echo '<td>'.$row['reg_time'].'</td>';
    echo '</tr>';
}
echo '</table>';
echo '<script>var lastId = '. $lastId .'</script>'; // remember the last id
?>
Javascript loaded at the same page:
$(document).ready(function() {
    var usersTable = $(".users");
    setInterval(function() {
        $.ajax({
            url: 'get_users.php?id='+lastId,
            dataType: 'json',
            success: function(json) {
                if (json.length > 0) {
                    $.each(json, function(key, user) {
                        usersTable.append('<tr><td>'+user.name+'</td><td>'+user.reg_time+'</td></tr>');
                        lastId = user.id;
                    });
                }
            }
        });
    }, 10000);
});
get_users.php:
$lastId = (int)$_GET['id'];
$query = mysql_query('SELECT * FROM `register` WHERE `id` >' . $lastId);
$result = array();
while ($row = mysql_fetch_array($query)) {
    $result[] = $row;
}
echo json_encode($result);
The main point is to make ajax call every X seconds and append all users that registered after the last user on the current page.
To read: