A new tile appears in a random empty cell: a 2 nine times in ten, or a 4 one time in ten. A move that changes nothing adds no tile.
After a move, one new tile is placed in a random empty cell. It is a 2 about 90% of the time and a 4 about 10% of the time. A new game starts with two tiles already on the 4×4 board.
Only moves that change the board
If you press a direction and no tile can slide or merge, nothing happens: no new tile, no score and no move used. That makes it safe to try a direction that cannot do anything, but it also means you cannot pass a turn.
Undo and new tiles
If you undo a move and play it again, the tile that appears is not guaranteed to land in the same place as before. The next tile is not replayed from the undone move. See how many moves you can undo. For where to keep your big tiles, read the strategy guide, or the full rules.