Jquery not loading / running on WordPress home page

You have initialized the script twice. Here is the updated script:

jQuery(document).ready(function($) {
var $window           = $(window),
win_height_padded = $window.height() * 1.1,
isTouch           = Modernizr.touch;

if (isTouch) { $('.revealOnScroll').addClass('.animated'); }

$window.on('scroll', revealOnScroll);

function revealOnScroll() {
    var scrolled = $window.scrollTop(),
    win_height_padded = $window.height() * 1.1;

    // Showed...
    $(".revealOnScroll:not(.animated)").each(function () {
        var $this     = $(this),
        offsetTop = $this.offset().top;

        if (scrolled + win_height_padded > offsetTop) {
            if ($this.data('timeout')) {
                window.setTimeout(function(){
                    $this.addClass('.animated ' + $this.data('animation'));
                }, parseInt($this.data('timeout'),10));
            } else {
                $this.addClass('.animated ' + $this.data('animation'));
            }
        }
    });
   // Hidden...
   $(".revealOnScroll.animated").each(function (index) {
       var $this     = $(this),
       offsetTop = $this.offset().top;
       if (scrolled + win_height_padded < offsetTop) {
           $(this).removeClass('.animated .fadeInUp .flipInX .lightSpeedIn')
       }
   });
}

revealOnScroll();
});

UPDATE:
This script is not working because you have used jQuery script inside Javascript which won’t work. You need to use either javascript or jQuery.

revealOnScroll() is a javascript function while .each used inside this function is a jQuery function. Please be clear whether you want to use jQuery or javascript.

UPDATE 1:

Add this script after jquery.js which you have missed.

<script src="http://ajax.aspnetcdn.com/ajax/modernizr/modernizr-2.7.2.js"></script>

Hope this helps!