Basic Craft/Yii Module + Controller

app.php

<?php

return [
    '*' => [
        'modules' => [
            'my-module' => \modules\Module::class
        ],
        'bootstrap' => [
            'my-module'
        ]
    ],
    // ... Probably more application config, here!
];

Module.php

<?php
namespace modules;

use Craft;

/**
 * Custom module class.
 *
 * This class will be available throughout the system via:
 * `Craft::$app->getModule('my-module')`.
 *
 * You can change its module ID ("my-module") to something else from
 * config/app.php.
 *
 * If you want the module to get loaded on every request, uncomment this line
 * in config/app.php:
 *
 *     'bootstrap' => ['my-module']
 *
 * Learn more about Yii module development in Yii's documentation:
 * http://www.yiiframework.com/doc-2.0/guide-structure-modules.html
 */
class Module extends \yii\base\Module
{
    /**
     * Initializes the module.
     */
    public function init()
    {
        // Set a @modules alias pointed to the modules/ directory
        Craft::setAlias('@modules', __DIR__);

        // Set the controllerNamespace based on whether this is a console or web request
        if (Craft::$app->getRequest()->getIsConsoleRequest()) {
            $this->controllerNamespace = 'modules\\console\\controllers';
        } else {
            $this->controllerNamespace = 'modules\\controllers';
        }

        parent::init();

        // Custom initialization code goes here...
    }
}

TestController.php

<?php

namespace modules\controllers;

use craft\web\Controller;

/**
 * This controller should be placed at `modules/controllers/TestController.php`!
 */
class TestController extends Controller
{
    /**
     * @var array Actions to allow anonymous (unauthenticated) access to.
     */
    public $allowAnonymous = ['my-test'];

    /**
     * Test action, accessible via `?action=my-module/test/my-test` (or with the `action` body param)
     */
    public function actionMyTest()
    {
        // Anything you need to do with the incoming data, including sending remote requests:
        // $myParam = Craft::$app->getRequest()->getBodyParam('myPostValue');

        return 'Hello, world!';
    }
}