Unable to save variable(s) from a custom form

Modify your example_form_form1_submit function as follows

function example_form_form1_submit(&$form, &$form_state) {
  variable_set('id_event', $form_state['values']['id_event']);
  $id_event = variable_get('id_event', '');
  drupal_set_message($id_event);
}

Drupal's variable API allows for storage of persistent variables across separate requests.

If you want to set a Drupal variable, you need to use variable_set().

Example:

$my_variable = 'very_important_value';          // Here we set a php variable

variable_set('my_variable_name', $my_variable); // Here we store the php variable
                                                // into a Drupal variable

If you want to retrieve the Drupal variable's value, you need to use variable_get().

$saved_value = variable_get('my_variable_name', 'default_value'); // Get the value of the
                                                                  // Drupal variable. If
                                                                  // it does not exist, then
                                                                  // use the second argument

You are seeing the value stored correctly in the database because you are making use of variable_set() correctly. However, when you are trying to retrieve the value, you need to be sure to call variable_get() to retrieve the value and store it into a php variable for further usage.

Suggestion:

function example_form_form1_submit(&$form, &$form_state) {

  variable_set('id_event', $form_state['values']['id_event']); // Stores the value

  $saved_value = variable_get('id_event', 'Default');          // Retrieve the stored value.
                                                               // If it is not saved,
                                                               // $saved_value will be set
                                                               // to 'Default'.
  drupal_set_message($saved_value);
}

I resolved my problem using "system_settings_form($form);" since the solutions above did not work for me. (Will try later and if it works, will post any learnings here).

Following is what worked for me for reference:

My .module file:

function neil_menu() {

$items['admin/settings/neil'] = array(
    'title' => 'Node images',
    'description' => 'Control how to upload node images.',
    'page callback' => 'drupal_get_form',
    'page arguments' => array('neil_admin_settings'),
    'access arguments' => array('administer site configuration'),
    'file' => 'neil.admin.inc',
);
 return $items;

}

neil.admin.inc file:

function neil_admin_settings() {
   $form['progress_so_far'] = array(
'#type' => 'textfield',
'#title' => t('Progress So Far'),
'#default_value' => variable_get('progress_so_far', '0'),
'#maxlength' => 255,
);

return system_settings_form($form);
}