Definition: A number of
coplanar
line segments, each connected end to end to form a closed shape.

Try this Drag the orange dots on each vertex
to reshape the polygon. It is initially an irregular polygon; change it by clicking on "make regular", and change the number of sides.

Polygons are one of the most all-encompassing shapes in geometry. From the simple triangle up through squares, rectangles,
trapezoids, to dodecagons and beyond.

In the figure above click on "make regular" and then gradually change the number of sides to see familiar shapes.

All
interior angles
less than 180°,and all
vertices
'point outwards' away from the interior. The opposite of concave.
Regular polygons are always convex.
See Convex Polygons

Concave

One or more interior angles greater than 180°. Some vertices push 'inwards'
towards the interior of the polygon. The opposite of convex.
See Concave Polygons

Self-intersecting or Crossed

A polygon where one or more sides crosses back over another side, creating multiple smaller polygons. Most of the properties and theorems concerning polygons do not apply to this shape.
It is best considered as several separate polygons.

A polygon that in not self-intersecting in this way is called a simple polygon.

Properties of all Polygons (regular and irregular)

The diagonals of a polygon are lines linking any two non-adjacent vertices.
See Diagonals of a Polygon

Area

For regular polygons there are various ways to calculate the area.
See Area of a Regular Polygon For irregular polygons things are a little harder since there no general formulae.
For a general approach see
Area of an Irregular Polygon

The apothem of a polygon is a line from the center to the
midpoint
of a side.
This is also the inradius - the radius of the incircle (see below).
See Apothem of a Polygon

The radius of a regular polygon is a line from the center to any vertex. It
is also the radius of the circumcircle of the polygon (see below).
See Radius of a Polygon

The incircle is the largest circle that will fit inside a regular polygon.
Its radius is the apothem of the polygon (see above).
See Incircle of a Polygon

The circle that passes through all the
vertices of a regular polygon.
Its radius is the radius of the polygon (see above).
See Circumcircle of a Polygon

Named Polygons

Many polygons have names based on the number of sides. A 5-sided polygon is called a pentagon for example.
There are some that wish to name every possible polygon, but there seems little point in doing so.
For example a 42-sided polygon is called a "tetracontakaidigon".

Beyond about 10 sides, most people call them an "n-gon". For example a 15-gon has 15 sides. This seems easier to remember and understand.
However, there are some names that do occur in everyday experience: