This videos defines the symmetric group and discusses the relationship of the symmetric group to the symmetries of an equilateral triangle and the symmetries of a square.
For those who are wondering why D_4 isn't the same thing as S_4: some permutations in S_4 are impossible to be reached just by applying the D_4 permutations. In the case of D_3 and S_3, all D_3 symmetries also belong to S_3. (Sorry for the bad English)