A cone has one vertex and one edge.
A cone has one vertex, one edge, and one base.
Two faces, one vertex and one edge on a finite cone.
A teardrop-shape would have one vertex (the tip), one edge and one face
it is a cone.
The shape that is commonly known as a cone has two faces, one vertex and one edge. Mathematically, however, the cone has infinite height and so has one [curved] face and one vertex.
It is called a cone.