Ajax not working es expected (Returns 0)

Please try this:

Add the following in your theme’s functions.php

add_action( 'init', 'tnc_aj_scripts' );

function tnc_aj_scripts() {
   wp_register_script( "ajax_anmeldung_script", get_template_directory_uri(). '/my-ajax.js', array('jquery'),  '1.0', true);
   wp_localize_script( 'ajax_anmeldung_script', 'mYAjax', array( 'ajaxurl' => admin_url( 'admin-ajax.php' )));        

   wp_enqueue_script( 'jquery' );
   wp_enqueue_script( 'ajax_anmeldung_script' );
}

add_action("wp_ajax_ajax_anmeldung", "ajax_anmeldung");
add_action("wp_ajax_nopriv_ajax_anmeldung", "ajax_anmeldung");

function ajax_anmeldung() {
    $result['one'] = 'Hey';
    $result = json_encode($result);
    echo $result;
    die();
}

Create a .js file in your theme directory called my-ajax.js and put the following inside that file:

jQuery('#teilnahme_form').submit(function(e){
    e.preventDefault();

    var name = document.getElementById("name").value;

    jQuery.ajax({
      type:"post",
      dataType : "json",
      url : mYAjax.ajaxurl,
      data: {
        action: "ajax_anmeldung",
        name : name
      },
      success:function(data){
        alert(data.one);
      }
   });
});