Nintendo Badge Arcade (JP) is an application for the 3DS to unlock special "badges" to decorate the home screen.
The game is primarily a crane game (though machines using hammers or bombs also show up from time to time.) The goal is to cause the badges to fall into the hole or holes that appear on the play area. The easiest way is to pick up the badge itself using the claw, but moving badges can cause other badges to topple, which may cause them to also fall, making it possible to win multiple badges with a single play. Playing the game costs $1 for 5 tries.
Free plays can be won in the daily practice game: Each day, you get 5 free tries on the practice game. You don't get to keep the badges won in the practice game, but for every 10 practice badges you collect, you win 1 free play in the real badge game. The number of badges carries over from day to day: if you collect 7 practice badges one day, the next day you would only need 3 more to get a free play. Clearing the practice machine with attempts left will refill it, allowing you to continue to play until you've used up the 5 free plays for the day.
In addition, the practice game sometimes contains special red and blue badges. The special status of these badges isn't shown while playing, but are revealed after your 5 attempts are used up. Collecting the red badge is worth 3 free plays in the real game, while a blue badge is worth 1.
The game has numerous badges to collect. Badges can be placed on the 3DS home screen to decorate it, or be placed on a folder to indicate what it contains. Some badges can replace the default icons for the default applications - clicking those badges will launch the associated application. Some badges unlock themes for the 3DS like a Mario Kart one. Badges are organized in series like Super Mario Bros., Animal Crossing and Mario Kart.
Every day, a badge layout is posted on the Miiverse Board. (add more info)