Well, for example are conjugate permutations in , since they both are the product of a 2-cycle and a 3-cycle, but none of them is conjugate to since this last is the product of two 2-cycles (transpositions), and etc.
Of course, one must know first that ANY permutation is expressable as the product of DISJOINT cycles...
If you want a geometric interpretation, use the fact that A4 is the group of rotations of a regular tetrahedron. The tetrahedron has three axes, namely the lines joining the midpoints of opposite edges. A rotation of the tetrahedron takes axes to axes, and thus induces a homomorphism from A4 to S3, whose kernel is the Klein group (consisting of the identity map together with rotations through 180º about each axis).