To extend laserlight idea: Have the current elements index in the a href and and identifier for up or down. i.e. <a href="reorder.php?index=1&direction='up">
Pseudo code:
If up:-
1) put the value of the selected item into a temp variable ie. $temp = $item[url.index]
2) Get the item above and set it to the index of the moved item. i.e. $item[url.index] = $item[url.index-1];
3) Get the last element in the Array and set it to the index-1. $item[url.index-1] = $temp
if down:-
1) $temp = $item[url.index]
2) $item[url.index] = $item[url.index+1];
3) $item[url.index+1] = $temp
You should put some checking in if they are trying to move the first or last item. Even if you hide the move up link for the first item or the down link for the last - its still good practise.