Definition: A polygon that has one or more
interior angles greater than 180°
(Result: some vertices point 'inwards', towards the the center.)

Try this
Adjust the polygon below by dragging any orange dot.
If any vertex points 'inward' to towards the interior of the polygon, it is a concave polygon.

A concave polygon is defined as a polygon with one or more
interior angles greater than 180°.
It looks sort of like a vertex has been 'pushed in' towards the inside of the polygon.
Note that a triangle (3-gon) can never be concave.

A concave polygon is the opposite of a convex polygon. See
Convex Polygon.

In the figure above, drag any of the vertices around with the mouse.
Take note of what it takes to make the polygon either convex or concave.
Also change the number of sides.

Properties of a Concave Polygon

A line drawn through a concave polygon, depending on exactly where you draw it, can intersect
the polygon in more than two places. In the figure on the right, the line cuts the polygon in 4 places.
You can also see that the line can divide the polygon into more than two pieces, here three.

Some of thediagonals
of a concave polygon will lie outside the polygon. In the figure on the right, the diagonal at the top of the polygon
is outside the polygon's interior space.
(In a
convex polygon,
all diagonals will lie inside the polygon).

The area of a concave polygon can be found by treating it as any other irregular polygon.
See Area of an Irregular Polygon

Regular Polygons are never concave by definition. See Regular Polygon Definition.
In the figure above, click on "make regular" to force the polygon to always be a regular polygon.
You will see then that, no matter what you do, it will remain convex, not concave.