Un modo per farlo è prendere prima la riga, quindi utilizzare un ciclo do/while invece di un semplice ciclo while. In questo modo:
if (mysql_num_rows($result)) { // Valid user ID, show the form.
/*********** I added this line ***********/
$row = mysql_fetch_array($result, MYSQL_NUM);
echo '<p>Order for:<strong>' . $row[2] . ' ' . $row[3] . ' </strong> </p>
<table border="0" style="font-size:11px;" cellspacing="1" cellpadding="5">
<tr class="top">
<td align="left"><b>Product</b></td>
<td align="center"><b>Price</b></td>
<td align="center"><b>Qty</b></td>
</tr>';
$bg = '#dddddd'; // Set the background color.
/*********** I changed this from a while loop to a do-while loop ***********/
do { // WHILE loop start
$bg = ($bg=='#eaeced' ? '#dddddd' : '#eaeced');
echo '<tr bgcolor="' . $bg . '">';
echo '<td align="left">' . $row[15] . '</td>
<td align="center">' . $row[13] . ' pts</td>
<td align="center">' . $row[12] . '</td>
</tr>';
echo '';
} while($row = mysql_fetch_array($result, MYSQL_NUM)); // end of WHILE loop