Pachinko's are essentially a Japanese equivalent of one-hand bandit machines originating from the USA. They're gambling machines. Gambling is banned in Japan, if I'm not mistaken, which is why these machines operate using small balls - you put some in at the top of the machine, they fall down passing through obstacles (akin to pinball tables) and you can win more of them to play again, or exchange them for money. During the play, animations, videos or music are being played on the screen to entertain you (just like pinball tables used graphics and pictures in their design) and display results or such.
The Wikipedia entry about pachinkos explains everything, from how they work to why are they somewhat controversial (and thus why would Ayu's collaboration with pachinko be controversial).
