Please let me know if you have already solved the problem.
If I understand your problem correctly, you have a list of Cigars which you want people to rate. When a user clicks on a cigar name, he is taken to another page to rate the cigar on different parameters.
Have you tried passing the cigar_id for a particular cigar through a get variable? <a href="rating_page.php?cigar_id=<cigar_id>">Cigar Name</a> Replace the red text above with the cigar_id number.
To pick up the variable on the next page, use the $Cigar_ID=$_GET['cigar_id']; statement.
Also, Instead of using so many queries to the database each time to get each rating, you can use just one SELECT statement like $sql="SELECT AVG(Rat_App) as a, AVG(Rat_Draw) as b, AVG(Rat_Flavor) as c, AVG(Rat_Burn) as d, AVG(Rat_Finish) as e, AVG(Rat_Overall) as f FROM Cigar_Reviews WHERE Cigar_ID = '$Cigar_ID'" .
You can then pick up the ratings of each using
$result=mysql_query($sql);
$row=mysql_fetch_array($result);
$a=$row['a']; $b=$row['b']; $c=$row['c']; $d=$row['d']; $e=$row['e']; $f=$row['f'];
and then do the required calculations.
Please let me know if this works or if any more help is required.