JavaScript moving element in the DOM

There's no need to use a library for such a trivial task:

var divs = document.getElementsByTagName("div");   // order: first, second, third
divs[2].parentNode.insertBefore(divs[2], divs[0]); // order: third, first, second
divs[2].parentNode.insertBefore(divs[2], divs[1]); // order: third, second, first

This takes account of the fact that getElementsByTagName returns a live NodeList that is automatically updated to reflect the order of the elements in the DOM as they are manipulated.

You could also use:

var divs = document.getElementsByTagName("div");   // order: first, second, third
divs[0].parentNode.appendChild(divs[0]);           // order: second, third, first
divs[1].parentNode.insertBefore(divs[0], divs[1]); // order: third, second, first

and there are various other possible permutations, if you feel like experimenting:

divs[0].parentNode.appendChild(divs[0].parentNode.replaceChild(divs[2], divs[0]));

for example :-)

Trivial with jQuery

$('#div1').insertAfter('#div3');
$('#div3').insertBefore('#div2');

If you want to do it repeatedly, you'll need to use different selectors since the divs will retain their ids as they are moved around.

$(function() {
    setInterval( function() {
        $('div:first').insertAfter($('div').eq(2));
        $('div').eq(1).insertBefore('div:first');
    }, 3000 );
});