Slide all tiles in one direction at a time. When two tiles with the same number touch, they merge into one tile worth double. A 2 and a 2 become a 4; two 4s become an 8, and so on.
Every move spawns a new 2 or 4 somewhere on the board. The goal is to keep merging upward until you reach 2048. If the board fills up and nothing can merge, the game ends.
Tip: keep your biggest tile pinned in one corner and build down from it. Don't let big numbers get stranded in the middle.