there are lot of issue in your Module>Please check

My config file in app/etc/module/dixit_featured.xml Path should be Dixit_Featured.xml from

dixit_featured.xml

and code

<?xml version="1.0"?>
 <config>
  <modules>
     <Dixit_Featured> <!--  Dixit_Featured to  dixit_featured -->
    <active>true</active>
    <codepool>local</codepool>
      </Dixit_Featured> <!--  Dixit_Featured to  dixit_featured -->
  </modules>
 </config>

Created Controllrer in app\code\local\dixit\featured\controllers\IndexController.php Path should be app\code\local\Dixit\Featured\controllers\IndexController.php

<?php
class Dixit_Featured_IndexController extends Mage_Core_Controller_Front_Action
{
   public function indexAction ()
   {
    echo 'test index';
   }
   public function mamethodeAction ()
   {
    echo 'test mymethod';
    }
}
?>

Class Should be dixit_featured_IndexController from Dixit_Featured_IndexController

And Created My Plugin xml file in app\code\local\dixit\featured\etc\config.xml to change xml file in app\code\local\Dixit\Featured\etc\config.xml and look like

<?xml version="1.0"?>
 <config>
    <modules>
       <dixit_featured><!--  Dixit_Featured to  dixit_featured -->
      <version>1.0.1</version>
       </dixit_featured> <!--  Dixit_Featured to  dixit_featured -->
     </modules>
<frontend>
<routers>
     <mymodule>
    <use>standard</use>
    <args>
        <module>Dixit_Featured</module> <!--  Dixit_Featured to  dixit_featured -->
        <frontname>dixit</frontname>
    </args>
     <mymodule>
</routers>
</frontend>
</config>

Please study basic study of module

there are lot of issue in your Module>Please check

My config file in app/etc/module/dixit_featured.xml Path should be Dixit_Featured.xml from

dixit_featured.xml

and code

<?xml version="1.0"?>
 <config>
  <modules>
     <Dixit_Featured> <!--  Dixit_Featured from  dixit_featured -->
    <active>true</active>
    <codepool>local</codepool>
      </Dixit_Featured> <!--  Dixit_Featured from dixit_featured -->
  </modules>
 </config>

Created Controllrer in app\code\local\dixit\featured\controllers\IndexController.php Path should be app\code\local\Dixit\Featured\controllers\IndexController.php

<?php
class Dixit_Featured_IndexController extends Mage_Core_Controller_Front_Action
{
   public function indexAction ()
   {
    echo 'test index';
   }
   public function mamethodeAction ()
   {
    echo 'test mymethod';
    }
}
?>

Class Should be dixit_featured_IndexController from Dixit_Featured_IndexController

And Created My Plugin xml file in app\code\local\dixit\featured\etc\config.xml to change xml file in app\code\local\Dixit\Featured\etc\config.xml and look like

<?xml version="1.0"?>
 <config>
    <modules>
       <dixit_featured><!--  Dixit_Featured from dixit_featured -->
      <version>1.0.1</version>
       </dixit_featured> <!--  Dixit_Featured from dixit_featured -->
     </modules>
<frontend>
<routers>
     <mymodule>
    <use>standard</use>
    <args>
        <module>Dixit_Featured</module> <!--  Dixit_Featured to  dixit_featured -->
        <frontname>dixit</frontname>
    </args>
     <mymodule>
</routers>
</frontend>
</config>

Please study basic study of module