Drupal - How to validate and submit a form using AJAX?

I know this is question has been lying around for a couple of years by now, but I don't feel any of the other answers really get the point of Drupal 7 ajax form submission, so I'll try to explain it.

Since your form should work as well without the ajax as per good practice, your ajax submit handler should do nothing but return the form. Everything else should be in your validation and submit functions.

function ex_form($form, $form_state) {
  $form['name'] = array(
    '#type' => 'textfield',
     '#title' => t('Name'),
     '#default_value' => '',
     '#maxlength' => '128',
     '#required' => TRUE,
  );

  $form['submit'] = array(
   '#type' => 'submit',
   '#value' => 'Submit',
   '#ajax' => array(
     'callback' => 'ex_form_ajax_handler',
     'effect' => 'fade',
   ),
  );
}

function ex_form_submit(&$form, &$form_state) {
  // Submit logic - runs when validation is ok
  // You can alter the form here to be just a markup field if you want.
}

function ex_form_validate(&$form, &$form_state) {
  // Validation logic - invalidate using form_set_error() or form_error()
}

function ex_form_ajax_handler(&$form, &$form_state) {
  return $form;
}

I think the post by maxtorete on October 17, 2011 seems to give a fuller example using both form_validate() and form_submit()

(I haven't tested it yet.)

Also Joshua Stewardson's answer over at stack overflow has a nice working example:

function dr_search_test_form($form, &$fstate) {

  $form['wrapper'] = [
    '#markup' => '<div id="test-ajax"></div>',
  ];

  $form['name'] = [
    '#type'     => 'textfield',
    '#required' => TRUE,
    '#title'    => 'Name',
  ];

  $form['submit'] = [
    '#type'  => 'submit',
    '#value' => 'Send',
    '#ajax'  => [
      'callback' => 'dr_search_test_form_callback',
      'wrapper'  => 'test-ajax',
      'effect'   => 'fade',
    ],
  ];

  return $form;
}

function dr_search_test_form_callback($form, &$fstate) {

  return '<div id="test-ajax">Wrapper Div</div>';
}

function dr_search_test_form_validate($form, &$fstate) {

  form_set_error('name', 'Some error to display.');
}

Joshua makes the point that validation error messages replace the #ajax['wrapper'] element completely so your callback needs to resupply this element again as you replace it.

In general, validation and form submission handling should happen in the usual _validate() and _submit() callbacks. The idea is that forms should still work without ajax.

Pretty much the only thing the #ajax callbacks should do is return the part of the form that should be replaced, according to the defined wrapper.