Login programmatically using ajax
Instead of using jQuery AJAX, you can achieve this by Drupal Form API AJAX.

You can hide the Actual submit button in the hook_form_alter using hide() and add a custom button with AJAX callback. 

Or if you still want to practice this using jQuery AJAX, Drupal's way of handling AJAX calls is little different. You may have to first understand the hook_menu in Drupal. Once you have a clear understanding of this system, you have to create pages using this hook. In drupal, it is not advisable to create custom php files for handling AJAX callbacks. Instead you can create a page using hook_menu and handle the server callback there. For the JS part, the Drupal way is to use Drupal Behaviors

This is a small example of how it would look like.

Your custom module to handle AJAX login (sign_up.module)

&lt;?php
function sign_up_menu {
    $items = array();
    $items['custom/user-login'] = array(
        'page callback' =&gt; 'custom_user_login_callback',
        'type' =&gt; MENU_CALLBACK,
        'access callback' =&gt; TRUE
    );
    return $items;
}
function custom_user_login_callback(){
    /**
     *  You can validate the User credentials here and provide error message for invalid credentials or send him to welcome page using drupal_goto() for valid credentials
    **/
}


-

The custom JS file (sign_up.js)

Drupal.behaviors.customSignUp = {
    attach: function(context, settings) {
        $("#test", context).once(function(){
            $(this).click(function(){
                $.ajax({
                      url: Drupal.settings.basePath + 'custom/user-login',
                      type: 'POST',
                      data: {
                        username: /** Pass the User Name **/
                        password: /** Pass the Password **/
                      },
                      success: function(response) {
                        $('#ajax-response-wrapper').html(response);
                        Drupal.attachBehaviors();
                      }
                });
            });
        });
    }
};