You can sign up using email and password, or using one of the supported social logins.
The application is what gets a subscription to an API, not the developer. Thus you need to register an application first. This is the API client.
From the APIs page, select the API you need for your application. Select a plan, and then subscribe to the API. This will give you credentials for accessing the API.
With the provided credentials, you may now use the API from your application according to the plan you subscribed for.