What does edges and vertices have in common?

Answer 1

Not much, other than they both can be parts of a 3-dimensional figure.

"Vertices" is plural of "vertex", which is a point. The vertex of an angle, for example, is the point of intersection between two lines. 2-dimensional and 3-dimensional figures can have vertices.

An "edge", however, is a line segment, and only 3-dimensional figures (cubes, pyramids, etc) can have edges. An edge is the intersection of two sides of a figure.