For each iteration of your
while ($row = mysql_fetch_assoc($shipping))
loop, $row is an associative array containing the field names and values that you selected in your query. So we have this on the first iteration: $row['id'] is '5' and $row['name'] is 'Other'. On the second iteration $row['id'] is '9' and $row['name'] is 'UPS'...
So what you want to do is check to see if what the user entered is equal to the checkbox that we're printing out at this iteration of the loop. If it is, you want to add the "selected" attribute to that checkbox.
I just don't know where your $REQUEST['shipping'] value is coming from. (By the way, the preferred method is to use either $POST['shippping'] or $_GET['shipping'].)
So to answer your question, that loop will spit out checkboxes that look like this:
<input type='checkbox' name ='shipping[]' value='Other'>Other
<input type='checkbox' name ='shipping[]' value='UPS'>UPS
...
If the form was submitted, $_POST['shipping'] will contain an array of all the checkboxes checked. This brings up another issue. I would assume that only one of these should be checked. If this is the case you should use radio buttons rather than checkboxes. I'm assuming you understand the difference.
Does this clear things up a little for you? Need more explanation?