contact form in template with jquery,validate and ajax

WordPress is running Jquery in noconflict mode. (WordPress Codex Reference)
or read this brief article I posted some time ago. View it here.

You need to replace:

$(document).ready(function(){

With this:

jQuery(document).ready(function($){

So all together, try this:

jQuery(document).ready(function($){
    $("#contact_form").validate({
        rules: {
            name: "required",
            email: {
                required: true,
                email: true
            },
            message: "required"
        },
        messages: {
            name: "Please enter your name",
            email: {
                required: "Please enter your email adress",
                email: "Please enter your valid email adress"
            },
            message: "Please enter a short message, what your inquiry is about"
        },
        submitHandler: function(form) {
            $('#loading_icon').show();
            $('#click').hide();
            var params = $(form).serialize();
            $.ajax ({
                type: 'POST',
                url: '<?php echo admin_url( 'admin-ajax.php' ); ?>',
                data: params + "&action=sendmail",
                success: function(response) {
                    $('#response').hide();
                    $('#response').html(response);
                    $('#response').fadeIn('slow');
                    $('#loading_icon').hide();                      
                }
            });
        }

    });
});