Elementor – How to create widget category of elementor on wordpress

As like the documentation: Getting Started for extending Elementor

Create a custom plugin first, hope you know How to create a custom plugin: Create plugins in wordpress

A short demo:
Remember full Elementor core is written in OOP.
Assuming the plugin name as ElementorTest

<?php
/**
 * Plugin Name: ElementorTest
 * Plugin URI:  https://example.com/plugins/ElementorTest/
 * Description: Basic Elementor Extension
 * Version:     1.1
 * Author:      Ahmed Maruf
 * Author URI:  https://author.example.com/
 * License:     GPL2
 * License URI: https://www.gnu.org/licenses/gpl-2.0.html
 * Text Domain: elementortestplugin
 * Domain Path: /languages
 */

Now, we need to make sure that no direct access is given to our plugin php files, and let us create the basic skeleton for creating an extension/widget via Elementor.Ref: Initialize the plugin

<?php
if ( ! defined( 'ABSPATH' ) ) {
    exit; // Exit if accessed directly.
}



final class Elementor_Test_Extension {

        const VERSION = "1.1"; //Your plugin version
        const MINIMUM_ELEMENTOR_VERSION = "2.0.0"; //Minimum Elementor Version Required
        const MINIMUM_PHP_VERSION = "7.0"; //Minimum PHP version required to run your plugin

        private static $_instance = null;
        /*The plugin class should use a singleton design pattern to make sure it loads only once*/
        public static function instance() {
            if ( is_null( self::$_instance ) ) {
             self::$_instance = new self();
         }return self::$_instance;

     }
     /*

      The constructor should initiate the plugin. The init process should check for basic requirements and then then run the plugin logic. Note that If one of the basic plugin requirements fails the plugin logic won’t run.
      */
      public function __construct() {
        add_action( 'plugins_loaded', [ $this, 'init' ] );
    }
    /*Initialize all the basic requirements to run the plugin logic*/
    public function init() {
        load_plugin_textdomain( 'elementortestplugin' );

        // Check if Elementor installed and activated
        if ( ! did_action( 'elementor/loaded' ) ) {
            add_action( 'admin_notices', [ $this, 'admin_notice_missing_main_plugin' ] );
            return;
        }

        // Check for required Elementor version
        if ( ! version_compare( ELEMENTOR_VERSION, self::MINIMUM_ELEMENTOR_VERSION, '>=' ) ) {
            add_action( 'admin_notices', [ $this, 'admin_notice_minimum_elementor_version' ] );
            return;
        }

        // Check for required PHP version
        if ( version_compare( PHP_VERSION, self::MINIMUM_PHP_VERSION, '<' ) ) {
            add_action( 'admin_notices', [ $this, 'admin_notice_minimum_php_version' ] );
            return;
        }

        // Add Plugin actions when rest requirements are passed
        add_action( 'elementor/widgets/widgets_registered', [ $this, 'init_widgets' ] );     

    }
    /*Callback function for the action hook admin notices*/
    public function admin_notice_missing_main_plugin() {

        if ( isset( $_GET['activate'] ) ) unset( $_GET['activate'] );

        $message = sprintf(
            /* translators: 1: Plugin name 2: Elementor */
            esc_html__( '"%1$s" requires "%2$s" to be installed and activated.', 'elementortestplugin' ),
            '<strong>' . esc_html__( 'Elementor Test Extension', 'elementortestplugin' ) . '</strong>',
            '<strong>' . esc_html__( 'Elementor', 'elementortestplugin' ) . '</strong>'
        );

        printf( '<div class="notice notice-warning is-dismissible"><p>%1$s</p></div>', $message );

    }

    /*Callback function for action hook admin notices upon elementor version not matching*/
    public function admin_notice_minimum_elementor_version() {

        if ( isset( $_GET['activate'] ) ) unset( $_GET['activate'] );

        $message = sprintf(
            /* translators: 1: Plugin name 2: Elementor 3: Required Elementor version */
            esc_html__( '"%1$s" requires "%2$s" version %3$s or greater.', 'elementortestplugin' ),
            '<strong>' . esc_html__( 'Elementor Test Extension', 'elementortestplugin' ) . '</strong>',
            '<strong>' . esc_html__( 'Elementor', 'elementortestplugin' ) . '</strong>',
            self::MINIMUM_ELEMENTOR_VERSION
        );

        printf( '<div class="notice notice-warning is-dismissible"><p>%1$s</p></div>', $message );

    }
    /*Callback function for action hood admin notices upon php version not matched*/
    public function admin_notice_minimum_php_version() {

        if ( isset( $_GET['activate'] ) ) unset( $_GET['activate'] );

        $message = sprintf(
            /* translators: 1: Plugin name 2: PHP 3: Required PHP version */
            esc_html__( '"%1$s" requires "%2$s" version %3$s or greater.', 'elementortestplugin' ),
            '<strong>' . esc_html__( 'Elementor Test Extension', 'elementortestplugin' ) . '</strong>',
            '<strong>' . esc_html__( 'PHP', 'elementortestplugin' ) . '</strong>',
            self::MINIMUM_PHP_VERSION
        );

        printf( '<div class="notice notice-warning is-dismissible"><p>%1$s</p></div>', $message );

    }

    /*
    @Callback function for the action hook elementor/widgets/widgets_registered
    @Create the folder widgets and the file under you custom plugin /widgets/test-widget.php
    */
    public function init_widgets() {

        // Include Widget files
        require_once( __DIR__ . '/widgets/test-widget.php' );

        // Register widget by creating the class in the file you have created naming as test-widget.php
        ElementorPlugin::instance()->widgets_manager->register_widget_type( new Elementor_Test_Widget() );

    }

    public function includes() {}

}
Elementor_Test_Extension::instance();

Now create a class under youtplugin/widgets/test-widget.php. Ref: Creating the core widget functionality

class Elementor_Test_Widget extends ElementorWidget_Base
{
    /**
     * Define your core logic for the widget
     */
    public function __construct()
    {

    }
}

Rest are pretty self explanatory. For widget category the skeleton is also similar. Hope this will help.