How to build an ajax form on Shopify. See http://inside.sauce.ly/how-to-build-an-ajax-login-form-on-shopify/ for the full blog post.

enforce-http.liquid.js

/*
* Ensure the http protocol is always used on the myshopify.com domains. 
* Uses liquid to input the correct URL.
*/
if (window.location.href.match(/https:\/\/.*.myshopify.com/) && top === self) {
  window.location.href = window.location.href.replace(/https:\/\/.*.myshopify.com/, 'http://{{ shop.domain }}');
}

shopify-ajax-login.js

/*
* Try to login, check the login credentials, and then redirect if required.
*/
login(email, password).done(function (html) {  
  if (html.indexOf('Invalid login credentials') !== -1) {
    // invalid password - show a message to the user
  } else {
    
    // All good! Let's redirect if required
    var checkoutUrl = getCheckoutUrl();
    if (checkoutUrl) {
      window.location.href = checkoutUrl;
    } else {
      // don't need to redirect, do what you like! 
    }
    
  }
});

/*
* Or if you're registering a user:
*/
register(email, password, firstName, lastName).done(function (html) {
  
  // logic for registration errors
  if (html.indexOf('email is invalid') !== -1) {
     
  } else if html.indexOf('email can't be blank') !== -1) { 
    
  } else if (html.indexOf('password can't be blank.') !== -1) {
      
  }

});

/*
* Log the customer in with their email and password.
*/
function login(email, password) {  
  var data = {
    'customer[email]': email,
    'customer[password]': password,
    form_type: 'customer_login',
    utf8: '✓'
  };

  var promise = $.ajax({
    url: '/account/login',
    method: 'post',
    data: data,
    dataType: 'html',
    async: true
  });

  return promise;
}

/*
* Log the customer in with their email and password.
*/
function register(email, password, firstName, lastName) {  
  var data = {
    'customer[email]': email,
    'customer[password]': password,
    'customer[first_name]': firstName,
    'customer[last_name]': lastName,
    form_type: 'create_customer',
    utf8: '✓'
  };

  var promise = $.ajax({
    url: '/account',
    method: 'post',
    data: data,
    dataType: 'html',
    async: true
  });

  return promise;
}

/*
* Get the `checkout_url` from the URL query parameter, if it exists.
* (It only ever exists on the /account/login page)
*/
function getCheckoutUrl() {  
  function getParameterByName(name) {
    name = name.replace(/[\[]/, "\\[").replace(/[\]]/, "\\]");
    var regex = new RegExp("[\\?&]" + name + "=([^&#]*)"),
        results = regex.exec(location.search);
    return results === null ? "" : decodeURIComponent(results[1].replace(/\+/g, " "));
  }
  return getParameterByName('checkout_url');
}