The Palm Jumeirah is a series of man made islands that when seen from the sky look like a giant palm tree. The massive building project was started in 2001, and is still on going. Presently there are many house and neighborhoods, as well as several hotels and a water park. There is a monorail that cuts through the center of the palm that will take you all the way from the city to the far end where many of the attractions can be found.
At this point from a tourist perspective the palm doesn't have much to offer, unless you're staying at one of the hotels actually on the palms. For the most part the hotels are fairly closed off to tourists and you will not have access to any of the beaches or gardens unless you're a guest. You do have access to the stores in the hotels, and in the Atlantis hotel there is an aquarium.
The main attraction for tourists on the palm is the Aquaventure Water Park. It's located next to the Atlantis hotel, and offers many fun water slides and fun areas for the whole family. As all things in Dubai the price of the water park can vary depending on the exspirences you want to have, starting price is around 250 AED, and can go up into the thousands for added features.
While there may be a great future for the Palm Jumeirah, including a mall located right in the middle, right now there's not really much to do unless you're a guest at one of the many hotels.