Whispers are a feature that allows for non-public discussion inside a public topic.
Before Whispers can be used, they need to be enabled by an admin by adding a group to whispers allowed groups in site settings. Once enabled, all members of those allowed groups can create and view Whisper posts.
Whispers can only be set when creating a post. To make a post a Whisper you can either select the 
icon in the composer toolbar and click Toggle Whisper:
Or set it from the Reply Toggle menu in the top left corner:
And the composer will indicate that the reply will be a Whisper:
Replying to a Whisper will also automatically bring up the composer with Whisper toggled on.
Unless you have the Toggle Whisper 👁 theme component enabled on your site, Whispers are permanent. There is no way to convert a Whisper to a normal post, nor is there a way to convert a normal post to a Whisper. If you need to do so delete the Whisper and re-post.