Where are the best cheap but absolutely delicious restaurants in DC?

I would consider the following good, cheap-ish options.

Julia's Empanadas - Empanadas... Obviously

Toki Underground - Ramen

TaKorean Union Market - Korean Tacos

Also, if you are in DC on the weekend, I think brunch at Founding Farmers is very reasonable and amazing.

I really enjoyed Busboys and Poets! They have two locations in DC but I've only been to the one at 5th Street a couple of times. Great atmosphere, tasty food, decent prices. :) You'll find their menu at http://www.busboysandpoets.com. 

All these places are great and I would agree that 14th St NWand U Street is a great place to look.  Some good places that are not super expensive but good are Masa 14 and El Centro D.F. for brunch or happy hour.  ChurchKeyhas great beer and food to match.  If the day is nice, try the backyard of Vinoteca Wine Bar and Bistro

If you are there in the afternoon and want great Greek food, try Greek Deli & Catering.  There are no seats, so be prepared to wait for a seat outside.  Or walk to the park

Bub and Pop's sandwiches near Dupont Circle. Great philly-style sandwiches in a central area.

Ben's Chili Bowl is a MUST. Great food and a long legacy. &pizza is also cheap and delicious! 

My favorite places are a little out of the way in DC but for a true cheap eats Jamaican food check out Jam Doung Style Cuisine on North Capitol.  It's a very small carry out place so be prepare to eat outside or take it to go.  For amazing Szechuan style chinese food, I recommend Panda Gourmet.  There is also a pretty good food truck scene in DC and you can usually find a great selection of trucks parked around the various parks downtown. 

For a delicious and cheap breakfast, go get pancakes at Eastern Market. You'll find many things to tempt you there, but the pancakes are a local rave.

Though it's not super cheap, the newer Rose's Luxury is really special. The food is unique and delicious and it's just one of those happy places. If you feel like splurging just a little, go there. (Get there early, it's very popular.) rosesluxury.com/

There are some great options in DC for breakfast, lunch and dinner. Bistro Bohem and Habesha Market And Carry-out (if you're adventurous). For lunch and dinner, I would explore the 14th and U Street, NW area. There are some great restaurants, lounges, and cafés in that neighborhood. 

Ben's Chili Bowl is legendary!