Each merge adds the value of the tile it creates to your score. Merging two 8s makes a 16 and adds 16 points. Sliding adds nothing.
Your score only changes when two tiles merge. The merge adds the value of the new tile: two 2s make a 4 and add 4 points, two 8s make a 16 and add 16, and two 512s make a 1024 and add 1,024.
What does not score
Sliding tiles without merging is worth nothing, and a new tile appearing is worth nothing either. Two merges in one move add both values together.
Your best score
If you undo a move, the score from before that move comes back with it. Your personal best is kept even if you undo. Because bigger merges are worth more, the corner method that builds one big tile also builds the score. See when the game ends and the full 2048 rules.