The Claw is a chance game, where you have to spend five gems to play it for each go. You can use your arrow keys to move the claw around, and press 'DROP' to see if you can get a prize. It is randomized, and it is definitely not a skill or strategy game. You can get plushies and items by winning The Claw, and if you own your own Claw you can get a Phantom Plushie from it, however that is very rare.
There are a few places The Claw is - in the Sarepia Theater, in the museum in Appondale, in the Captain Melville's Juice Hut and many more other locations. They were once given out to Jammers as monthly gifts. It is rare if you have one of your own Claws, but you could probably buy one with diamonds at the diamond shop.
Most of The Claw mini games share the same plushie variety, which is virtually all the plushies in the game except the Puffer Fish plushie which you get as a reward for collecting all the prizes in a area, and the Phantom Plushies, which can only be won in the den version of The Claw game. However, some parties have them, and not all The Claw games just give Plushies, they can also give items. For example, in the Trading Party, there is The Claw machine which gives you cool items, sometimes you can even get rare items!