Not far from the White House is a restaurant called Ten Penh (for both Ten Pennsylvania Avenue and for the capital of Cambodia). It is pan-Asian and a local favorite. Not cheap, but memorable--as you requested.\

If you can get to Georgetown, 1789 is excellent continental (also not cheap). More moderate is the Vietnamese restaurant (also in Georgetown) Miss Saigon (their curry chicken is wonderful). And if you are looking for cuisine you might not find elsewhere (but plentiful in DC), try Meskerem in Adams-Morgan for Ethiopian (also moderate).

You might want to check out the Washingtonian magazine for its list of the 100 best restaurants (also the 100 best bargain restaurants) in Washington or Tom Sietsema's reviews in the Washington Post .

Hope that this helps.