The goal of the API is to provide communication between Form Tools and the outside world. It contains functionality for tasks such as submitting form submissions via PHP code rather than POSTing in the information to process.php, accessing and display Form Tools data in your own webpages, managing client accounts through code rather than the Form Tools interface and other such helpful functionality.
The Form Tools API files should be placed in your [form tools root]/global/api/ folder.
API v1.x vs API v2.x
Form Tools 3 was a complete rewrite of the application, changing the codebase from functional code to object-oriented. The API v2.x was updated for Form Tools 3 compatibility, converting the API into a class.
In terms of functionality, v1.x and v2.0.0 of the API are identical: they (currently) provide the same methods. What's changed has the way you call the API methods.