Swipe and every tile slides as far as it can go. Two of the same number that meet become one of the next.
One swipe slides every tile on the board, not just one. They travel until they hit a wall or another tile — there are no half-moves and nothing stops in the middle.
Arrow keys or WASD on a keyboard.
Two 2s make a 4, two 4s make an 8, and so on. The number you make is added to your score, so a big merge is worth far more than a small one.
A row of four 2s becomes two 4s — not a single 8. Anything made by a merge is finished for that move and will not merge again until the next one.
This is the rule that makes the game hard. Without it a board would collapse in on itself and never fill up.
A new tile appears after every move that changes something. When the board is full and no two neighbours match, that is the game.
A swipe that would change nothing is not a move — it is refused, and nothing new appears. You can never fill the board by swiping at a wall.
Reaching it is celebrated once and then you carry straight on — the board is left exactly as it was. There are milestones at 128, 256, 512, 1024, 2048 and 4096, and each is recorded the first time you ever get there.
Undo takes back one move, and only one. It covers a swipe that went the wrong way; it deliberately will not let you re-roll the tile that appeared, which would be a different and much easier game.