How do I check a form is a node type form?
In the case you are referring to the node edit form (the form that allows you to edit a node), instead of hook_form_alter(), you need to implement hook_form_BASE_FORM_ID_alter(), which is a specific implementation of hook_form_alter() used when you know the form ID of the base form. In your case, you need to implement hook_form_node_form_alter() exactly how the Book module does with book_form_node_form_alter().
function book_form_node_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  $node = $form['#node'];

  // Omissis.
}

$form['#node'] is set by node_form(), which contains the following code. (Note the comment, which says that $form['#node'] is going to be removed in Drupal 8, and that modules should instead use $form_state['node'].)
// @todo D8: Remove. Modules should access the node using $form_state['node'].
$form['#node'] = $node;

In the case you are referring to the form you find at admin/structure/types/manage and that allows you to edit the properties of a content type, you need to implement hook_form_FORM_ID_alter(), which a specific implementation of hook_form_alter() done for a specific form ID, which in this case is node_type. This is what does the Comment module with comment_form_node_type_form_alter(). If you need to know the content type for which the form is shown, you can use $form['#node_type']-&gt;type.