Vertices, Edges and Faces
A vertex is a corner.
An edge is a line segment between faces.
A face is a single flat surface.
Let us look more closely at each of those:
Vertices
A vertex (plural: vertices) is a point where two or more line segments meet.
It is a Corner.
This tetrahedron has 4 vertices.
And this pentagon has 5 vertices: 
Edges
This Pentagon
Has 5 Edges
For a polygon an edge is a line segment on the boundary joining one vertex (corner point) to another.
This Tetrahedron
Has 6 Edges
For a polyhedron an edge is a line segment where two faces meet.
Faces
A face is any of the individual flat surfaces of a solid object. This tetrahedron has 4 faces (there is one face you can't see)

Sides
"Side" is not a very accurate word, because it can mean:
 An edge of a polygon, or
 A face of a polyhedron
Euler's Formula
For many solid shapes the
 Number of Faces
 plus the Number of Vertices
 minus the Number of Edges
always equals 2
This can be written: F + V − E = 2
Try it on the cube: A cube has 6 Faces, 8 Vertices, and 12 Edges, so: 6 + 8 − 12 = 2 (To find out more about this read Euler's Formula.) 