When you left-click on a link you can drag it to a new position on the same desktop. You can also move it to a different desktop in the same Album by dragging it to the desktop tab and then releasing the finger.
To Move a link to a different desktop in a different Album, right-click the link and choose "Copy this link", and then go to the destination album/desktop and select a position there, right click and choose "Paste Copied Link".
On a tablet, you tap the link and choose "Move on desktop", and then tap on a different position on the desktop and the link should move to that position. To copy to other desktops, tap and Copy the link, or a group of links individually (separately) and then go to the other desktop and tap and choose from the menu "Place selected links here" - which will move them to there - or "Copy selected links here", which copies them.
Copying or Moving a Group of links together: Select the group, either individually or by moving your cursor over the group. When all links are selected, place the cursor near the group, right-click and choose Copy or Move selected group. A new dialog will then instruct you how to copy or move (Place) them to a different desktop or album.
You can "Align" links on a desktop, stacking them neatly relative to each other, in order to improve the overall appearance of the desktop. To do this, first select the links you wish to align together, either by right-click/Select Link individually, or by holding down the left mouse button and dragging the cursor from above and slightly to the left of the group to be aligned to the right and down over the whole group. Each link will then be selected and you can right-click beside any one of the selected group and choose "Align selected group" and pick spacing options if needed.
Later you can right-click near the (still selected, now aligned) group and choose "Deselect all". You can then move one or more links out of alignment if you wish. (While all the links - the group - are still selected, you can hold down the cursor/mouse on any one of the links and drag the group to a new position.)
On tablets, tap the link instead of right-click, then use the context menus as described above.
To save space on a desktop, yet stay within your links quota (if any) you can of course delete a link. But if you don't want to totally lose it, you can "archive" it. The archive is a section of your database where links can be stored, but not activated from there. Nevertheless, they can be restored to the original desktop if needed. To do this, just right-click (or tap in a tablet) the link and "Move to Archive". Later you can find the archives, for each desktop which has one, under the Admin menu/ Links Archive, and you can click to Restore any link from there.