Wasat Al Balad
It may look dirty, sketchy, and chaotic, and indeed it is dirty, sketchy, and chaotic, but downtown Amman, also known as Wasat Al Balad (center of the nation), is the beating heart of the city. It's easy to get lost in the winding and hilly streets, lined with shops selling just about everything. While most of Amman is peaceful and orderly, downtown is awash in humanity. Despite the shabby look, it's actually quite safe. As in the rest of the city, you don't have to worry too much about pickpockets and scammers.
There's a fruit and vegetable souk located behind the King Hussein Mosque. There's also a gold souk between the Mosque and the Citadel selling all sorts of jewelry. Hashems is Jordan's most famous restaurant. Located in a tiny alley on a side street, it's hard to find but worth the trouble. Hashems only sells falafel, hummus, and a few other traditional Arab staples, but it executes them perfectly and cheaply. There are no shortage of street food vendors if you're in the mood for kababs, shawarma, or a fresh juice.
Wasat Al Balad is also the place to go for souvenirs. You won't want to leave Jordan without a traditional keffiyeh. You can find all sorts of antiques and knickknacks too. There are loads of knockoff goods -- fake designer bags, luggage, and pirated DVDs. Prices on most everything are negotiable. Remember to bargain!
Getting to downtown is easy. Just hail a taxi and ask for Wasat Al Balad. They'll drop you off right in the center, near the King Hussein Mosque. Spend an afternoon here wandering around and taking it all in. You can't say you've been to Amman without doing so.