The Joomla order column makes saving the order of items easy...

Clicking the icon will save the current state of the order column.

To move the item in spot 6 to spot 1, replace 6 with 1 and click .

Don't worry if there's already an item assigned to spot 1.

If the item does not go to the proper spot (it might be one place out) use the up/down arrows to nudge it into position. 

The order column can be used to reorder all items at once. Enter the new values and click the disk icon.