The API is a programming interface. It includes a particular set of rules ("code") and specifications that software can follow to communicate with each other. For Drupal, there is the API reference that shows the documentation available to developers. There is also a "Form API" and a "Field API".
Aegir is a distribution that automates many of the common tasks related to deploying, maintaining, or backing up your Drupal sites. It is a "super distribution" in the sense that it allows to administer all the distributions installed on your server. Do you want to deploy a DrupalCommerce, OpenAtrium, OpenEntreprise, Drupal7, Drupal8 site? Your deployment takes less than 5 minutes if you master your domains :) Moreover, the distribution allows you to manage more efficiently your multi-site architectures, or the administration of a site with heavy traffic in the case of a multi-server hosting.
AMOA is the Project Management Assistance, that is to say the bearer, the commander of a project. In the case of a Web project, its role is to help clarify the needs and explain it for the providers in charge of the project. In concrete terms, it organizes workshops with the client, drafts the specifications and / or the tender. It can also intervene during the recipe.