How to create dynamic select list in form using FAPI
You need to use the #ajax property in your State select list, with a callback function that returns the district element.  So something like this:

$form['state'] = array(
  '#title' =&gt; t('Name of your State:') ,
  '#type' =&gt; 'select',
  '#default_value' =&gt; array(
   ),
  '#options' =&gt; $options,
  '#required' =&gt; '1',
  '#ajax' =&gt; array(
    'callback' =&gt; 'MY_MODULE_state_ajax_callback',
    'wrapper' =&gt; 'edit-district',
 );


then create a callback called MY_MODULE_state_ajax_callback($form, &amp;$form_state)
and within the function, 

MY_MODULE_state_ajax_callback($form, &amp;$form_state) {
    $form_state['rebuild'] = TRUE;
    return $form['district'];
}


It may be slightly more complicated than this, based on your use-case, but this is the meat of it.