I have trouble understanding what exactly a dihedral group is. I read about how they are rotations and reflections along the faces of a polygon. But then what does that mean? Whatever you do to a polygon you get the same polygon right? Could someone give a dihedral group for dummies explanation?
And also what does it mean for a shape to have a certain dihedral symmetry?
Cut out a regular 5-gon from a paper sheet with a blade, and get 5-gon and paper sheet with the 5-gonal hole. Write your name on the 5-gon with blue pen on the one side, and with red pen on the other side. Then try to fit it back in the hole. How many different ways to do it? Two ways are different if your name have different color or different position. The answer to your question: each "way" is the element of the dihedral group $D_5$.
The final note is that $5$ may be any natural number.