A cycle is a permutation that, as its name suggests, moves certain pieces in a cycle. An n-cycle will affect certain pieces p1...pn so that p1 moves to the place of p2, p2 moves to the place of p3 (and so on...), pn-1 moves to the place of pn and pn moves back to the place of p1.
The following example illustrates a 3-cycle of edges:
In this 3-cycle, the blue piece moves to the yellow spot, the yellow piece moves to the green spot, and the green piece moves back to the blue spot.
Obviously, if a 3-cycle is performed 3 times in a row, then all pieces will return back to their original positions.
The last step of the Heise method usually involves solving a 3-cycle of corners such as the one below:
Such 3-cycles can be intuitively constructed using commutators.