Greenwich Village is lower Manhattan's artsy, bohemian neighborhood, especially popular with the Beatniks in the 1950s (Allen Ginsberg, Jack Kerouac, etc.) and folk singing hippies in the 1960s (Joan Baez, Bob Dylan). Check out its many fine alleys.
Less bohemian now, but still buzzing, Greenwich Village is home to many excellent restaurants, music clubs, art galleries and sex shops.
For an idea of the 1960s vibe, listen to The Roches talk about Folk City, the legendary Greenwich Village music club, and sing their song Face Down at Folk City.