SoClassy did a decent job explaining this. Still a few little details missing though.
When you post an answer, question or comment, if you edit it again in the five minutes after it's posted, you can edit it without an "edited by" message appearing on it. The edits will also be invisible in the edit history. Any user can do this.
Note that if you start editing the post in those five minutes but publish it after the five minutes are gone, the "edited by" message will show up. However, I believe this does cause a glitch where the link to the edit history doesn't show up on the "edited by" message, and still won't show up if you edit it again. This doesn't actually affect the actual edit history page though, which you can access by adding "revisions" and a forward slash before the post number in the URL.
As for silent editing, only mods get access to that. This will also prevent the "edited by" message from appearing (and will stop it from updating if it's already there), but it can't stop the edits from being added to the edit history, which you can access via the method explained.
Hopefully that answers your questions!