Reply To: What do you think of that hand?
those ‘centers’ are an Overlay and can be enabled here:
They are the calculated average of the Vertices that make the Face and therefore can even lie outside of the Face as you can see.
With Catmull-Clark, (in simple cases like this) each Edge of a Face is subdivided once to make new Vertices and each original Vertex shifts halfway towards that ‘center point’. The complete algorithm is slightly more complicated, but this might already help understanding some of the behavior of the Subsurf Modifier.