According to the Euler characteristic, for a simply connected solid,
V - E + F = 2 where
V = number of vertices (corners), E = number of edges and F = number of faces.
If E = 8 and V = 8, as required by the question, then F = 2.
There is no simply connected solid with just two faces.
Octogon. Eight sides, think of OCTOpus which has eight legs.
solid with 2 edges is a cylinder
A sphere is a solid with no edges.
A pyramid.
sphere
square-based pyramid
Square pyramid
It is a triangular prism.
With eight corners, it must be an octagon.
4 corners
A vertex.
That is a vertex.