hook_form_alter() is the go-to function for making changes to forms that are used across Drupal. We all use it. We all love it.

But since Drupal 7, you can also use a variant of this function: hook_form_BASE_FORM_ID_alter(). Here's a simple visual of how it works: 

hook_form_alter() fires on all Drupal forms (the big circle). With hook_form_BASE_FORM_ID_alter(), you can restrict it to firing on a subset of forms (one of the smaller circles), which can save Drupal some processing. (to get even more specific, you could target a single form by using hook_form_FORM_ID_alter(), but I'll leave that discussion for another day).

You can find the Base Form ID by printing out the $form_state variable in a normal hook_form_alter, like so:

function mymodule_form_alter(&$form, &$form_state, $form_id) {
// You must enable the devel module in order to use this function.
dpm($form_state); // Prints $form_state to the page.
}

The "base_form_id" is tucked away in the "build_info" array.

If you want to implement hook_BASE_FORM_id_alter, here is a simple example:

function mymodule_form_node_form_alter(&$form, &$form_state, $form_id) {
// Find the content type of the node we are editing.
$content_type = $form['#node']->type;
if ($content_type == 'article') {
// Alter the 'article' form here.
}
}

As you can imagine, these tools aren't worth much if you don't know which Base Form ID's are out there. I dug around Drupal core a bit and these are the ones I was able to find.

Base Form IDs in Drupal 7:

• node_form --- for node edit forms
• comment_form --- for comment forms

Base Form IDs in Drupal 8

(as of August 16, 2013)

• node_form --- for node edit forms
• comment_form --- for comment forms
• menu_form --- for menu forms
• menu_link_form --- for menu link forms
• node_type_form --- for content type forms
• custom_block_form --- for custom block forms
• taxonomy_term_form --- for taxonomy term forms
• contact_category_form --- for site-wide contact category forms
• date_format_form --- for date/time format forms
• image_style_form --- for image style forms
• filter_format_form --- for text format forms
• view_form --- for views forms
• custom_block_type_form --- for block type forms (new in Drupal 8)
• form_mode_form --- for form mode forms (new in Drupal 8)
• view_mode_form --- for view mode forms (new in Drupal 8)

Base Form IDs in major Contrib modules:

• webform_client_form --- for webforms (from the webform module)

As you can see, the API is implemented most thoroughly in Drupal 8. If you know about ones that I missed, just mention them in the comments below and I'll add them to the list.