The Dead Sea is a unique environment. It is so named because nothing can live in the hyper-saline water, which fluctuates around 31.5% salt. The Dead Sea is also the lowest point in the world -- its surface 423m below sea level. Because of decreased inflow from the Jordan River, the Dead Sea is slowly shrinking. The entire southern half has become disconnected and is now maintained only for industry. Plans are in the works for a canal connecting the Red Sea and the Dead Sea, which would refill it and address Jordan's chronic water issues. It would also permanently alter the Dead Sea's unique chemistry.
No trip to Jordan is complete without a trip to the Dead Sea. You're of course going to want to take a dip, or more accurately a float. Be careful though: if you have any cuts, the water can be excruciating. You'll also want to stay on your back, as it can be irritating if it gets in any orifices. The water and mud is considered therapeutic and the coast is dotted with spas and resorts. It rarely rains, and is generally 5C to 10C warmer than Amman, so it's an especially good place to visit in the winter.
The Dead Sea is located about 50km southwest of Amman, and well connected by road. The main beach is Amman Beach, close to the northwest corner. Buses regularly leave from Abdali Station, and are cheap and convenient. You can hire a private taxi to take you for around 20JD. The resorts also offer shuttles.