In the following set of figures, only b, d, and e are polygons:

Notice that the polygons only contain (straight) segments, these segments do not overlap, and the endpoints of segments are always joined.  An important property of any polygon is that it can be completely filled with non-overlapping triangles with their vertices at the vertices of the polygon.

In view of this, we can define a polygon as follows:

DEFINITION:     A polygon is a two-dimensional figure consisting of at least 3 vertices (points) and at least 3 sides (straight segments) such that:

1)            each vertex is the endpoint of exactly 2 sides

2)            each pair of vertices are the endpoints of exactly one side

3)            if 2 sides intersect then their intersection is a vertex.

Now let's test this definition to see if it covers everything about polygons and nothing more.  First consider the three polygons:

The vertices are the endpoints of the sides.  In b there are 12 vertices, in d there are 10, and in e there are 4.  They are the "corners" where the sides meet.  Clearly these three figures satisfy the definition above.  (By the way, vertices is plural of vertex.)

Now consider the figures that are not polygons:

Figure a is not a polygon because vertices A and C are not the endpoints of the same side.  Figure c is not a polygon because vertex E is the endpoint of more than two sides (sides  .)  Figure f is not a polygon because side  is not a segment (straight).