Postal Service, The District Sleeps Alone Tonight
DC is a transitory place. Many people come here for jobs and internships, put in their time and then go back to where they come from. This Postal Service song nicely captures that loneliness and dislocation--and how the brightly lit, but deserted, downtown feels after dark.
Chuck Brown, Block Party
Until recently, DC hasn't been known for its hip-hop, but you can't talk about the city without mentioning go-go. Go-go is the legendary local music subculture (with the eponymous boots) that never quite hit the bigtime. It's basically a descendant of '70s funk but with a unique backbeat to it. Chuck Brown is the guy generally credited with creating the sound.
Junk Yard Band (Live 1994)
The Junk Yard Band was apparently formed in the '80s by a bunch of 10-year-old kids from the Barry Farms projects banging on hubcaps and plastic tubs.
Fugazi, Turnover (Live 1991)
The other DC music subculture that never hit it big (mostly by design) was the DC hardcore scene of which Fugazi is probably the most widely revered.
The Magnetic Fields, Washington D.C.
I'm partial to this Magnetic Fields song, but I was only able to find this cover version (which is kinda cute anyway but lacks the cheerleader shouting of the original).
Duke Ellington, Take the 'A' Train
Ellington was born in D.C. in 1899 and launched his career here.
-- update! --
Speaking of hip-hop, Wale is the local MC who's been getting some attention lately. He's got an album out next month and this single ft. Lady Gaga. Decent. Plus he's got a lot of DC pride. (No embedding, but you can listen here.)