What is the difference between flush mount, recessed, and zero clearance fireplace installation?

Let's start with recessed (or built-in) installation. This is when you create a niche in a wall and insert the back of the fireplace in it.

Flush mount installation is a type of recessed installation when the frame of the fireplace sits right on top of the wall surface (flush) without any gaps while the back of the fireplace is recessed into a wall.

And zero clearance term is usually used in reference to fireplace inserts. It means that they can be installed directly against combustible materials like wood, walls, or paneling. Basically, zero clearance inserts can be flush mounted.