See how to edit points.
A point is simply a point in space as defined by four numbers (X, Y, Z, W).
A vertex is a reference to a point. Primitives use vertices to reference points. For example, the corners of a polygon, the center of a sphere, or a control vertex of a spline curve.
Primitives can share points, while vertices are unique to a primitive.
For example, you can have polygons that have corners at an identical location in space. The polygons may share a single point, in which case each polygon’s vertex at that corner would be a reference to the same point. Or, those vertices may each point to unique points that happen to have the same spacial location.
This is not merely an academic distinction. Uniqueing the points (giving each vertex its own point instead of sharing points at shared corners) with the Facet SOP has several important effects on polygons:

Because OpenGL no longer computes shared point normals, the polygons are individually shaded with a flat look.
However, you can precompute the normals to preserve the smooth shading of the original surface.

You can independently scale the polygon faces to create interesting effects.
If you precompute the normals , the individual pieces retain their shading from the original surface.
Similarly to uniqueing all points, you can use the Facet SOP to cusp polygons, uniqueing points selectively based on the angle at which the shared edges meet.