2 places I recommend
These are the two restaurants in Amman I have eaten at myself and would recommend.
A Jordanian restaurant on Rainbow Street, in Jabal Amman. Really delicious food — it almost felt home-cooked.
Address: Rainbow Street 26, Jabal Amman, Amman, Jordan
A restaurant on Taha Hussein Street, in Jabal Amman. The best Lebanese food I've ever had and the best kibbeh nayah ever (please note I've never been to Lebanon).
Address: Taha Hussein Street, Jabal Amman, Amman, Jordan