Well, take an element of D_6, and compose it with itself until you get the identity.
Each of the rotations has order 3 (if 6 = 2n)
Each of the reflections has order 2.
Dihedral Group D6 -- from Wolfram MathWorld
In the article, x is a rotation of 60 degrees. So....
x*x is a rotation of 120 degrees.
x*x*x is 180 degrees
x^6 is 360 degrees = 0 degrees, so x^6 is the identity. Being the smallest such exponent (for x), SIX is the order of x.
If you repflect (the operation "y" in the article), then reflect again, you get back to where you started.
What is your understanding of the order of an element in a Group (in general) ? Methinks there is something lacking.