The official documentation is at: http://docs.alfresco.com
What is the Alfresco API
The Alfresco API is a public API that enables you to persist content to Alfresco in the Cloud. The API consists of two parts: CMIS, for creating, querying, updating, and deleting documents and folders, and Alfresco REST calls for Alfresco functionality that CMIS doesn't cover.
Applications that use the Alfresco API run on behalf of Alfresco Cloud users who grant applications permission to use their account through an OAuth2 handshake.
How Do I Sign Up?
Signing up is easy, just go to http://developer.alfresco.com and sign-up. You can add as many applications to your account as you wish.
The Alfresco API documentation is available on the Alfresco Developer Portal here.
There are a few simple examples showing how to make Alfresco API calls:
Java command-line examples. These examples show how to make basic calls against the Alfresco API using Google's OAuth2 client and Apache Chemistry OpenCMIS. There are also some screencasts that show this code in action.
Java web application example. Gethin James has put together this web application.
Grails example. Peter Monks has a simple Grails app.