A cycle is a permutation that, as its name suggests, moves certain pieces in a cycle. An ncycle will affect certain pieces p_{1}...p_{n} so that p_{1} moves to the place of p_{2}, p_{2} moves to the place of p_{3} (and so on...), p_{n1} moves to the place of p_{n} and p_{n} moves back to the place of p_{1}.
The following example illustrates a 3cycle of edges:

In this 3cycle, 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 3cycle 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 3cycle of corners such as the one below:
Such 3cycles can be intuitively constructed using commutators.