The rough geography
Manhattan runs north to south, and locals navigate by "uptown" (north), "downtown" (south), and "crosstown" (east-west) rather than compass directions. Below 14th Street, the grid breaks down into older, angled streets from before the 1811 city plan — this is why Greenwich Village and the Financial District are so easy to get turned around in, while Midtown's numbered grid is nearly impossible to get lost in.
Downtown (below 14th St)
The Financial District, Tribeca, SoHo, Greenwich Village, the East Village, Chinatown, the Lower East Side, Meatpacking District, Tribeca, and SoHo. Older buildings, narrower streets, and the neighborhoods most associated with "New York character" in film — cobblestones, cast-iron facades, small bars. Also home to the 9/11 Memorial, the Brooklyn Bridge's Manhattan side, and some of the city's best-regarded restaurants.
Midtown (roughly 14th–59th St)
The skyscraper core: Times Square, the Empire State Building, Broadway's theater district, Grand Central, Fifth Avenue shopping. The most "postcard" version of the city, and priced accordingly for food and hotels.
Uptown (59th St and north)
Central Park splits into the Upper West Side and Upper East Side, both largely residential with major museums (the Met, the Natural History Museum, the Guggenheim). Further north, Harlem carries deep jazz and civil rights history alongside a fast-changing food scene, and Washington Heights/Inwood are quieter, more residential, and less visited by tourists than they probably should be — Inwood Hill Park alone is worth the A train ride.