Press U or R' in the turn panel and that face turns. Pick a sequence in the sequence panel and it starts from the solved cube and repeats, counting the rounds until the cube comes back. Drag the cube to look at it from another angle, use the wheel to move closer, and double-click to face it again. 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 turns of a Rubik's Cube form a group. Do one turn and then another and the result is again a turn of the cube, 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.
The six face turns are the generators. However tangled the cube looks, that state was built out of those six moves, and it can be taken apart with them as well. The inverse turns are not extra: U' is only U done three times.
Pick a sequence and repeat it and the cube always returns to solved. The number of rounds it takes is the order of that element. U alone comes back in 4, and R U, which is only two faces alternating, takes 105. Nothing about the cube makes this happen: in any finite group, repeating one element has to come back to the start eventually. The longest any sequence can take here is 1260 rounds, and R U2 D' B D' is one of them.
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. They do interfere, so a little is left over, and how little is the point: R U R' D R U' R' D' moves only three pieces. Being able to disturb a small part without stirring the rest is what makes solving methods possible at all.
The group has 43,252,003,274,489,856,000 elements. Eight corners can be arranged in 8! ways and twelve edges in 12!, the corner twists give 3 to the 7th and the edge flips 2 to the 11th, and the product is halved. The exponents are 7 and 11 rather than 8 and 12 because the twists have to sum to a multiple of three and the flips have to come in pairs. The final halving is the parity rule: corners and edges are always permuted with the same parity. Take a cube apart and put it back together at random and only one arrangement in twelve can be solved.
Shuffle applies twenty turns chosen at random. There is no solver here on purpose. What this page is for is not how to solve a cube but what the moves are: elements of a finite group, each with an order, each undoable, and all of them built from six turns.