The most common Arabic/Jordanian sweets are probably Kunafeh, Baklawa, Warabat, and Ma’amoul of all kinds. There is probably much more and it is usually really good.
Kunafeh is a mix of cheese, sweet fried rice noodles, sugar syrup, and mixed nuts. The best Kunafeh is to be found at Jabri, Habiba, Nafisa, or Sahel Al-Akhdar. All of them have several locations. Kunafeh should absolutely be fresh and eaten hot. Then you can enjoy it the most. The most well-known location for Kunafeh is probably Habibah in Downtown or Nafisa near the 7th circle.
Baklawa in the Middle East is very tasty. It is usually a mix of nuts baked in puff pastry with butter and sweetened with sugar syrup. It comes in many different styles, usually depending on the used nuts, such as walnuts, peanuts, almonds, pistachios, and cashew nuts. What’s inside also defines the price. Also there different shapes and many times you can find a mixed box, sold by kilos. In Jordan, you can find it in many places, even in Carrefour. It is a bit cheaper there, but there is a better place to look out for it. Zalatimo Brothers for Sweets (my favorite) or Jabri and Habiba sell really good Baklawa. Also, the Baklawa here usually tastes much better than in Turkey as it is not drowned in sugar as much.
Warabat is also a sweet based on puff pastry but unlike Baklawa not stuffed with nuts but with cream or cheese. It is a bit lighter, not so expensive, and also best to be consumed when it’s fresh and hot. However, you can also freeze it and defrost it and heat it up and it still will be nice. My favorite is from Jabri.
Ma3moul is a pastry based on semolina and is usually stuffed with dates, pistachios, or nuts. It is a tasty treat and comes in different sizes and styles. You will definitively not come around this. My favorite Ma3moul is from Sahel Al Akhdar. They sell Warabat and Baklawa as well. All the sweets come in nice metal boxes so they are great for gifts or something to take home when visiting your family.
Zalatimo Brothers for Sweets have several branches across Amman while their headquarters are in Shmesani. They offer the best collection of Baklawa, Warabat, and Cakes in town. You also find chocolate, honey, syrups, and other tasty treats there. In Ramadan, they also make some special sweets. You have to try their Baklawa and the awesome cakes they make. The Baklawa comes in a huge variety with all kinds of different nuts and shapes. They make mixed boxes for you on request, always fresh, and put it in a nice metal box suitable for travel. Baklawa can be kept well for about three months, so you don’t need to worry about expiration. You can also put it in the freezer if you intend to keep it even longer, but of course, it best to be enjoyed when fresh. Regarding the cakes, you can tell them about any kind of theme, the size or amount of people and they will create a personal cake. I got an awesome cake for my birthday from my wife. So I had to come up with something, too.
You can find decent ice cream for example at Rawan Cake or Gerard Ice Cream. Both of them have several locations throughout Amman. If you are looking for a somewhat new experience and crazy flavors you might enjoy a visit to Four Winters. They make ice cream using liquid nitrogen and create some really interesting flavors, such as the Bees Knees with Honey and Chili or Kunafeh. They have two branches, one on Abdali Boulevard and one in Abdoun. The ice is really creamy, smooth, and tastes awesome. That, however, has its price.
Update April 2018 – You can enjoy a similar experience at Nitros Ice Cream near Safeway 7th Circle. If the experience and variety of flavors are not enough for you, you might try your ice-cold sweets in the bubble waffle drowned in chocolate.