$db_link = коннект к бд.
$query = mysql_query("SELECT id, name, description FROM categories ORDER By id DESC", $db_link);

$content = '<ul>';
while($row = mysql_fetch_array($query)) {
$content .= '<li><a href="?action=archive&categoryId='.$row['id'].'">'.$row['name'].'</a></li>';
}
$content .= '</ul>';

echo $content;

<?php

$dbhostname = 'mysql.hostinger.ru';
      $dbusername = 'u4910715353_12';
      $dbpassword = 'password1';
      $dbname = 'u4910715353_21';

$db_link =  mysql_connect($dbhostname, $dbusername, $dbpassword);
$query = mysql_query("SELECT id, name, description FROM categories ORDER By id DESC", $db_link);   

$content = '<ul>';   
while($row = mysql_fetch_array($query)) {   
$content .= '<li><a href="?action=archive&categoryId='.$row['id'].'">'.$row['name'].'</a></li>';   
}   
$content .= '</ul>';   

echo $content;   
?>

// кол-во результатов
if (mysql_num_row($query)) {
// обрабатываем
}

<?php

$dbhostname = 'mysql.hostinger.ru';
     $dbusername = 'u49107211533512_12';
     $dbpassword = 'password1';
     $dbname = 'u4910715213512_1';

$db_link =  mysql_connect($dbhostname, $dbusername, $dbpassword);    

$query = '"SELECT id, name, description FROM categories  ORDER By id DESC".$db_link.';

$content = '<ul>';  
while($row = mysql_fetch_array($query)) {  
$content .= '<li><a href="?action=archive&categoryId='.$row['id'].'">'.$row['name'].'</a></li>';  
}  
$content .= '</ul>';  

echo $content;  
?>

$db_link =  mysql_connect($dbhostname, $dbusername, $dbpassword) or die('Mysql connect error');  
$query = mysql_query("SELECT id, name, description FROM categories ORDER By id DESC", $db_link) or die('Query error: '.mysql_error());