Magento – Calling Controller Path /foo/bar/baz Results in 404

Extending a Module is done in the same way as you would with core. If it is simply changing the nature you can override the class For Example : My_Mod_Model_Sales_Order extends There_Mod_Model_Sales_Order

Then just use the same method call so that your method is called instead of the purchased Mod.

You will to create a directory structure for you mod in "Local". The key is to ensure you have a rewrite in your config.xml

See http://www.magentocommerce.com/knowledge-base/entry/magento-for-dev-part-1-introduction-to-magento#9 For an good description of the reasons for the layout (He describes the logic behind the search that the code is doing) and the rest of the series for the good intro to the whole area.

For Example:

<global>
    <models>
        <sales>
            <rewrite>
                <order>My_Mod_Model_Sales_Order</order>
            </rewrite>
        </sales>
    </models>
</global>

If you can use an observer to capture events and to inject new code/data it is easier to manage and does not have the issue of other code stepping on your toes.

For controllers you need to do a little more work. For an example here is an overriding the CartController (to fix the problem of a case sensitive coupon...)

require_once(Mage::getModuleDir('controllers','Mage_Checkout').DS.'CartController.php');
class My_Checkout_CartController extends Mage_Checkout_CartController 
{
    public function couponPostAction()
    {.......}
}

And the associated config.xml

<config>
    <modules>
        <My_Checkout>
            <version>1.0.0</version>
        </My_Checkout>
    </modules>
    <frontend>
      <routers>
        <checkout>
          <use>standard</use>
          <args>
            <modules>
                <my_checkout before="Mage_Checkout">My_Checkout</my_checkout>
            </modules>
          </args>
        </checkout>
      </routers>
    </frontend>
</config>

You need some little modification.

Jurgis/Twilio/registration.php

\Magento\Framework\Component\ComponentRegistrar::register(
    \Magento\Framework\Component\ComponentRegistrar::MODULE,
    'Jurgis_Twilio',
    __DIR__
);

Jurgis/Twilio/etc/module.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Module/etc/module.xsd">
    <module name="Jurgis_Twilio" setup_version="2.0.0">
    </module>
</config>

Jurgis/Twilio/etc/frontend/routes.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:App/etc/routes.xsd">
    <router id="standard">
        <route id="Jurgis" frontName="Twilio_settings">
            <module name="Jurgis_Twilio" />
        </route>
    </router>
</config>

Jurgis/Twilio/Controller/Index/Display.php

namespace Jurgis\Twilio\Controller\Index;


class Display extends \Magento\Framework\App\Action\Action
{
    public function __construct(
        \Magento\Framework\App\Action\Context $context)
    {
        return parent::__construct($context);
    }

    public function execute()
    {
        echo 'Hello World';
        exit;
    }
}