As my Lonely Planet - Jordan tells me, one cheap and extremely delicious restaurant you should not miss in Amman is Hashem. I think it's actually called Hashem Cafe or Hashem Restaurant or Hashem Alley, but everybody just calls it Hashem. If you walk down the hill into east Amman from Rainbow St., Hashem is tucked away off the main road with all the shops on it. We actually got turned around while trying to find it (it really doesn't have a sign and is just some tables in an alley), but if you say Hashem to anybody, they'll point you in the right direction.
If you are worried about having digestive problems while eating in Jordan, Hashem is the ultimate test. With no utensils and no napkins, if you can survive eating at Hashem, you can eat any ol' thing from the side of the road in Jordan.
The servers all wear green long sleeve polo shirts as uniforms despite the stifling heat, and stare especially hard if you are a woman unaccompanied by a man, and even more so if your hair is uncovered. The service is really quick - there are no menus, and you can only choose among a handful of dishes.
From front to back: Foul (fool) madamas, a bean dip with olive oil; extra spicy chili sauce reminescent of my Sechuan grandmother's chili sauces; hummus drizzled with olive oil; small balls of falafel. The falafel at Hashem is really and definitively the best I have ever had, just like Din Tai Feng soup dumplings are definitively the best in the entire world. I had a dream later that night about ordering a giant bag of falafel and just having it in my purse to snack on while we toured around Jordan.
You also have the option of getting fries and large balls of falafel. The large ball of falafel has onions and other spices mixed in. We traded some small falafel for the big one because it was quite good. The proper procedure for eating at Hashem is to tear off a bit of bread, dip in foul or hummus, and eat it with a bit of tomato/onion/mint and a ball of falafel.
The great thing about Hashem is that it is cheap. We thought that with a full table of food it would be at least a few JD per person, but our total (includes all the dishes above, 3 bottles of water and hot tea) was only 4JD or so. To pay, you simply get up and walk to the platform in the middle of the alley, on top of which there is portly middle aged man sitting with a small desk. He is the owner, and also the one who takes your money. Expect to pay maybe a little more than 1JD per person (~$1.40) for a very filling meal.
Gratuitous night shot of east Amman