Players slide numbered tiles on a four-by-four grid to combine factors of three. For example, ones and twos merge to become a single "three" tile, two threes merge into "six", and two sixes merge into "12". Swiping the screen up, down, left, or right moves all of the tiles on the grid in that direction and adds a new tile to the grid in the same direction. The color of the incoming tile is shown onscreen. Players can preview moves by sliding the grid without letting go. Each kind of number tile has its their own personality, and new kinds of number tiles are introduced with a screen full of confetti when first unlocked.
Games of Threes typically last several minutes and end when no moves remain on the grid (usually when gridlocked with a single high number tile and many low number tiles). When a game is finished, there is no "game over" screen, but players receive a final score based on the rarity of the tiles (rather than the tile number values). The object of the game is to earn a high score. Outside of the game, players can review their scores and set Game Center challenges.
Source: Wikipedia, "Threes!", available under the CC-BY-SA License.