As of May 2017, there is no formal Forms API in the wordpress core.
There is the Fields API proposal that is working towards having a standard library for defining form fields.
The repo for the library can be found here:
Although this library is in an alpha state, and not recommended for production.
There is a discussion of the issues here:https://torquemag.io/2016/02/wordpress-needs-fields-api-use/
Now wordpress is moving towards a more REST API driven admin area, I think that this Fields API is an important step to standardising the Wordpress back end, which to be honest desperately needs it.
Until then the best option is to use one of the plugins mentioned in the other answers.