Press r or s in the turn panel and the polygon moves: r turns it by one vertex, r' turns it back, s flips it. Pick a sequence in the sequence panel and it starts from the numbered polygon and repeats, counting the rounds until the numbers come back. The display panel changes the number of sides, from a triangle up to a twelve-sided figure. The names at the top right open and close each panel, panels can be dragged anywhere and snap to each other, and double-clicking a panel title pins one panel to the top right instead.
The moves that carry a regular polygon onto itself form a group, called the dihedral group. Do one move and then another and the result is again such a move, there is a move that does nothing, every move has another move that undoes it, and when three moves follow each other it does not matter which pair you read first. That is the whole definition of a group, and here every line of it is something you can press.
Two moves are enough to build all of them. Turning by one vertex is r, flipping about the axis through vertex 0 is s, and those two are the generators. Turning back is not a third move: r' is r done n minus one times. For an n-sided figure there are n rotations and n flips, so the group has 2n elements, and the readout counts them for the shape you are looking at.
Pick a sequence and repeat it and the numbers always come back to where they started. The number of rounds it takes is the order of that element. Turning alone takes n rounds, so a hexagon comes back in 6 and a heptagon in 7, while any single flip comes back in 2. Nothing about polygons makes this happen: in any finite group, repeating one element has to come back to the start eventually.
Rotate then flip, and flip then rotate, are different moves. Both bring the figure back onto itself, both come back in two rounds, and yet they send vertex 1 to different places. Watch the numbers and you can see it. That is what it means for a group to be non-commutative, and it is the reason the order of the moves has to be written down and cannot be rearranged. A flip turns a rotation around: s r s is r' rather than r.
A commutator is a move, another move, the first one undone, the second one undone. If the two moves did not interfere with each other the four would cancel and nothing would happen. Here it is written r s r' s, because a flip undoes itself and needs no mark of its own. What it leaves behind is a rotation, r twice over, which is exactly how far the two moves fail to commute. On a triangle that rotation is a third of a turn, and running it three times brings the numbers home.
Take the rotations by themselves and you have a group inside the group, a subgroup, and it is the cyclic group: one move repeated n times. The flips are the other half, and every one of them is its own inverse. The same two halves appear again in the symmetry of tilings and crystals, and the way a Rubik's Cube behaves is this same reasoning on a larger group.