Wordpress - Does Wordpress have a "Form API"?

No, but it should ;)

There are several custom field class's (backend).

• wpAlchemy : https://github.com/farinspace/wpalchemy
• Meta Box Script: https://github.com/rilwis/meta-box
• My-Meta-Box: https://github.com/bainternet/My-Meta-Box
• meta-box-class: https://github.com/corycrowley/meta-box-class
• Meta Boxes Class: https://github.com/Bakke/Wordpress-Custom-Meta-Boxes-Class

For front-end forms, you probably best off with a plugin in Eugene Manuilov's link.

There is no Form API. Try to use plugin for it. Find it here: wordpress.org/extend/plugins/search.php?q=form

Another custom field library:

Custom Metaboxes and Fields for WordPress

https://github.com/jaredatch/Custom-Metaboxes-and-Fields-for-WordPress

A lot of the 'free' form plugins in the repo are very limited in functionality, but there are options:

GravityForms

http://www.gravityforms.com/

NinjaForms

http://wpninjas.net/plugins/ninja-forms/