How to make JQuery-AJAX request synchronous

From jQuery.ajax()

async Boolean
Default: true
By default, all requests are sent asynchronously (i.e. this is set to true by default). If you need synchronous requests, set this option to false.

So in your request, you must do async: false instead of async: "false".

Update:

The return value of ajaxSubmit is not the return value of the success: function(){...}. ajaxSubmit returns no value at all, which is equivalent to undefined, which in turn evaluates to true.

And that is the reason, why the form is always submitted and is independent of sending the request synchronous or not.

If you want to submit the form only, when the response is "Successful", you must return false from ajaxSubmit and then submit the form in the success function, as @halilb already suggested.

Something along these lines should work

function ajaxSubmit() {
    var password = $.trim($('#employee_password').val());
    $.ajax({
        type: "POST",
        url: "checkpass.php",
        data: "password="+password,
        success: function(response) {
            if(response == "Successful")
            {
                $('form').removeAttr('onsubmit'); // prevent endless loop
                $('form').submit();
            }
        }
    });

    return false;
}