Each request in this API documentation has got a "Try it out!" button that lets you try it's functionality.
The common part of this API contains public requests, that can be used without a login and private requests that require a login.
The exchange and wallet parts of this API contain only private requests that require that you log in before using them.
To use the private requests you first need to register. You can either register on the home page (click "Home") or you can use the "Try it out" function of /public/register in the common part of this API. After registering you must confirm your registration via the confirmation mail you receive.
After registering you can log in with your credentials either on the home page or with the "Try it out" function of the /public/authenticate request from the common part of this API.