First, add a field in your tables, like link_order, and if your tables are already filled, write some simple script that will set a sequence from 1 to n.
Second, when you query your links, order them by this link_order, DESC or ASC depending on what you want.
Make the MOVE UP link to point at, for example, move.php?id=$lid&dir=up, where $lid is the primary key of the link.
In move.php fetch id and dir variables, and use code not unlike this one:
$res= myqsl_query ("SELECT link_order FROM table WHERE id=$id LIMIT 1");
$old_order= mysql_result ($res, 0);
if ($dir=="up") $new_order= $old_order -1;
else $new_order= $old_order+1;
// Now swap the two links' positions:
$res= mysql_query ("SELECT id FROM table WHERE link_order=$new_order LIMIT 1");
$nid= mysql_result ($res, 0);
$res= mysql_query ("UPDATE table SET link_order=$old_order WHERE id=$nid LIMIT 1");
$res= mysql_query ("UPDATE table SET link_order=$new_order WHERE id=$id LIMIT 1");
Note that his is extremely basic example code and you need failsafe mechanisms, like checking if moving up has reached top. Likewise for moving down. Then if the input values are correct, etc... You migtht also wish to set an index to link_order