Form API, can you explain it in simple terms?

...so don't feel any chagrin about your initial confusion.

Have you looked at the resouces on http://api.drupal.org? In particular, start with this guide: http://api.drupal.org/api/4.7/file/developer/topics/forms_api.html

One of the useful things about he Forms API is that it defines an abstract representation of a form and then allows modules to alter the form before it is actually turned into HTML. This is frequently used in core- for example, to add the taxonomy selector to the node submssion form. It can also be handy for tweaking the behavior of Drupal's core modules- see, for example: http://drupal.org/node/70906

Create
If you want to make a form (i.e name, address, email), then do A, B, and C.

In a module (a file with a .module extension):

A. Write a function that creates a form:

function examplemodule_example_form() {
  // The form is an array. It doesn't need to have any direct properties.
  $form = array();
  // Each 'element' on the form (which can be a field) is an element in the form array. This first one we'll give the name 'title'.
  $form['title'] = array(
    // We give the form element a series of properties, which are keys to the element's array and have names starting with '#'.
    // One property is the type, which controls how the form element will be 'rendered' (output as HTML).
    '#type' => 'textfield',
    // We can give various other properties.
    '#title' => t('The title of your article'),
  );
  return $form;
}

B. Return and render the form in a function:

function examplemodule_example_page()  {
  $form = examplemodule_example_form();
  // The first argument becomes the 'form_id' of your form.
  $output = drupal_get_form('examplemodule_example_form', $form);
  return $output; 
}

C. Do something with the input of your form:

When someone fills out your form, Drupal will look for a function named [form_id]_submit.

function exampleform_example_form_submit($form_id, $form_values) {
  // E.g., do an SQL insert based on the $form_values['title'].
}

If you want to remove or add a form element (i.e. a collapsed field) from an existing form (i.e. create page), then do A, B, and C.

A. In your module, include a function called modulename_form_alter.

function examplemodule_form_alter($form_id, &$form) {
  // Look for the form you want to alter, by id:
  if ($form_id == 'othermodule_form') {
    // Add new form elements, or, as in this case, change existing ones. Because $form is passed 'by reference', changes you make here will be made to the form.
    $form['title']['#description'] = t('Choose a descriptive title.');
  }
}

There's no B or C.

Look for examples in other modules of functions named like _form_alter.

1. Module implementing form (navigate to ?q=exampleform):

function exampleform_menu($may_cache) {
  $items = array();
  if ($may_cache) {
    $items[] = array(
      'path' => 'exampleform',
      'callback' => 'exampleform_form',
      'title' => 'example form',
      'access' => user_access('access content'),
      'type' => MENU_CALLBACK
    );
  }
  return $items;
}

function exampleform_help($section) {
  switch ($section) {
    case 'admin/modules#description':
      return t('Example of form construction.');
  }
}

function exampleform_form() {
  $form['parent'] = array(
    '#tree' => TRUE
  );
  $form['parent']['1'] = array(
    '#type' => 'textfield'
  );
  $form['submit']  = array(
    '#type' => 'submit',
    '#value' => t('Submit')
  );
  return drupal_get_form('exampleform_form', $form);
}

function exampleform_form_submit($form_id, $form_values) {
  foreach ($form_values['parent'] as $key => $value) {
    drupal_set_message(t('Value: %value', array('%value' => $value)));
  }
}

2. Module doing a sample form alter:

/**
 * @file
 * Sample of form altering.
 */

/**
 * Implementation of hook_help().
 */
function example_help($section) {
  switch ($section) {
    case 'admin/modules#description':
      return t('Make a sample form alter.');
  }
}

/**
 * Implementation of hook_hform_alter().
 *
 * On node edit forms, alter the description of the title field.
 */
function example_form_alter($form_id, &$form) {
  // Match node editing forms. This function will affect all node editing forms but not any other form.
  // In general, you look for the 'form_id' of the form you want to alter. You can generally find it by
  // looking in the HTML source of the form as it's rendered on a page.
  // This test is taken from taxonomy_form_alter().
  if (isset($form['type']) && $form['type']['#value'] .'_node_form' == $form_id) {
    // Set a description, referencing the node type.
    if (isset($form['title'])) {
      $form['title']['#description'] = t('Please describe your %type.', array('%type' => node_get_name($form['type']['#value'])));
    }
  }
}