You need to unbind the click event from everything, not just the item being clicked. Instead of using $(this).unbind('click')
, you need to unbind it from all clickable elements, like this (this is an example just using the 2nd button, but it'd need to be done for both):
$(".appIMG2").one('click.appIMG2', function() {
$('.appIMG1, .appIMG2').unbind('click');
$("#app1").animate({
right: '250px',
opacity:0
});
$("#app3").fadeIn("slow");
});
});