The updateOrder() function contains examples of the custom class in action like:
$this->_db->setQuery( "UPDATE $this->_tbl"
  . "\nSET ordering='".$orders[$i]->ordering."' WHERE $k='".$orders[$i]->$k."'"
);
$this->_db->query();
Apart from being rather hard to read, we can see that the setQuery() and query() methods are nothing more than painfully crafted reimplementations of prepare() and execute() methods that have existed in most standard database APIs for decades, and which has been introduced to PHP is a standard API as PHP Data Objects (PDO).
Let's rewrite this using PDO:
$query = $this->_db->prepare("UPDATE $this->_tbl
                              SET ordering = ?
                              WHERE $this->_tbl_key = ?";
$query->execute(array($orders[$i]->ordering, $orders[$i]->$k));
While defining a basic database access abstraction layer for your application seems like reinventing the wheel when things like PDO, MDB2, ADODB, and PEAR::DB already exist, none of these were real options when FRED was being developed. Still, a cautionary tale for you: make sure you abstract the right layer of your application.