If you don't have a car, or do not want to make the trek to Korea Town in Annandale, Virginia, Mandu http://www.mandudc.com/ on 18th Street is a good substitute in the District. There is a shocking lack of Korean restaurants in DC. It's really a shame. Mandu's flavors are a little less spicy than what you'll find in Annandale, and you won't be able to cook your bulgogi on your table, but it's a great introduction to Korean food for the uninitiated. They also have a delicious selection of sojutinis.
Vegetable somen soup with carrots, mushrooms, bok choy and egg ($10)
Dolsot Bibimbap with tofu ($12)
Bibimbap is fantastic. Rice in a hot stone bowl is topped with carrots, mushrooms, spinach, tofu and a fried egg. Add in the desired amount of kochujang for spice and mix.
Duk Bok Gi ($11)
Thick rice cakes in a spicy chili paste with beef and onions.
Kimchee (part of the pan chan provided to each table)
Fried Vegetable Mandu ($5)
While the steamed version is healthier, I prefer the crispy texture.
Mandu has energetic servers and a menu that provides a nice introduction to Korean food. It's great for those times you end up with a large group with different dietary needs and you're looking for something affordable.