The Postmark Messages API allows developers to retrieve copies of messages sent through the Postmark API, providing a set of endpoints for querying and retrieving message details. The API offers various parameters for filtering messages by sender, recipient, tags, and subject, and returns an array of selected parts of the messages in descending order by sent date. To retrieve the full details of a message, developers can use the `details` endpoint with the MessageID. The Messages API is designed to work with both outbound and inbound posted messages, allowing for a workflow to search for bounce notifications and then retrieve the original message via the API.