The Maine Avenue Fish Market is a DC institution, drawing people from all walks of life. If you would like to sample fresh seafood at great prices, particularly the local specialty of Maryland crabs (and crabcakes!), this is the place to go. On weekends and especially in the summer, expect a large, diverse crowd and occasionally even live music.
The market, which is located right by the Potomac River near downtown, consists of a cluster of different vendors. Several serve cooked seafood that's often selected and cooked right in front of you. Don't expect an upscale experience - there are a few wooden tables but no proper seating and it smells like, well, a fish market.
Walk from the L'Enfant Plaza metro stop. It is recommended that you visit during the day only as the area can be dodgy at night.