conjugacy class
Inside a group, two elements are “the same up to a change of viewpoint” if one becomes the other after you relabel everything by some fixed symmetry. Conjugation, g -> x g x^{-1}, is exactly that relabeling. A conjugacy class gathers together all elements that look identical under some such change of coordinates.
Formally, elements a and b of a group G are conjugate if b = x a x^{-1} for some x in G. This is an equivalence relation, and its classes are the conjugacy classes; they partition G. Equivalently, the conjugacy class of a is the orbit of a under the action of G on itself by conjugation, so by orbit-stabilizer its size equals [G : C_G(a)], the index of the centralizer of a.
Conjugacy classes are coordinate-free invariants of an element: conjugate elements share order, and in a symmetric group conjugacy is detected purely by cycle type. Class functions — functions constant on conjugacy classes — are the natural functions to study a group through, and characters of representations are the prime example. The number of conjugacy classes of a finite group equals its number of irreducible complex representations.
In S_3 the conjugacy classes are exactly the cycle types: the identity {e}, the three transpositions {(1 2), (1 3), (2 3)}, and the two 3-cycles {(1 2 3), (1 3 2)}. Their sizes 1, 3, 2 sum to 6 = |S_3|, and there are 3 classes — matching S_3's three irreducible representations.
The three conjugacy classes of S_3, by cycle type.