Check out this straightforward example featuring navigation buttons.
HTML Content
<ul>
<li>1 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
<li>2 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
<li>3 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
<li>4 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
<li>5 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
<li>6 <a class='up' href='#'>move up</a> <a class='down' href='#'>move down</a></li>
</ul>
Interactive JavaScript using JQuery
$('.up').on('click', function(e) {
var container = $(this).closest('li')
container.insertBefore(container.prev())
})
$('.down').on('click', function(e) {
var container = $(this).closest('li')
container.insertAfter(container.next())
})
Experiment with it here on JSFiddle