Quickie, overly general Manhattan geography lesson:
First, as a comparative, uptown is North of where you are, downtown is South.
Second, the numbered streets start more-or-less at Houston (pronounced How-ston, not like the city in Texas), and proceed north from there.
Let’s start with Midtown, since that’s pretty well defined. Midtown is generally considered the area from about 34th Street to 59th Street. Since the Great Depression/World War II or so, Midtown has been the primary area for entertainment, tourism, hotels, retail commerce, the garment industry, and the media industry, among other things. In the later 20th Century, a large portion of the corporate, finance, legal, accounting and related industries has moved to Midtown from the financial district (discussed below). There is relatively little residental there. It is home to Macys, the Empire State Building, Rockefeller Center, Grand Central Terminal, Penn Station, the U.N., Times Square, “Fifth Avenue” shopping and countless other attractions.
Between 34th Street and 14th Street are several areas that are more residential and less built up, though there is substantial commerce. This section of town includes Chelsea, Murray Hill, Gramercy, and Flatiron, among other neighborhoods.
Below 14th Street is the area which can definitively be called downtown. From Houston to 14th Street is considered Greenwich Village, except for the far eastern portion (Alphabet City, i.e. Avenues A, B, C, and D), which is considered part of the Lower East Side. The Village is mostly residential, with some light retail commerce and entertainment, along with New York University.
South of Houston, North of Canal Street, and west of Broadway is considerd SoHo (South of Houston). Long ago it was an industrial area, but now it is almost entirely converted to residental lofts and galleries, with some light retail.
East of Broadway in this area is considered the Lower East Side, which has been home to countless waves of immigrant populations over the past several centuries, including Irish, German, Italian, Jewish, Chinese, Korean and many other groups. It includes Little Italy and Chinatown. With rising Manhattan property values, it is gentrifying into an attractive (possibly trendy) residental area for non-immigrants. (Indeed, virtually all of the immigrant or minority enclaves in Manhattan are gentrifying in Manhattan.)
Below Canal Street and West of Broadway is the area now known as TriBeCa (Triangle Below Canal). Like SoHo, this is a former industrial loft building area that is almost fully gentrified into residential.
Finally, we get to the area below City Hall. This is considered the Financial District, or “Downtown” in the more restrictive sense. The Financial District was, unsurprisingly, the home of the brokerage, banking and financial services industries until those businesses began moving up to midtown in the later 20th Century. Currently the financial services sector is pretty well split between Downtown and Midtown. Until the last 10 years or so, the Financial District was almost entirely non-residential, except for Battery Park City, a newish development on the Hudson River, west of the World Trade Center Site. Since then, there has been a considerable movement to convert commercial space into residential, and the Downtown population has increased substantially.
Moving to above 59th Street, we have Central Park dividing Manhattan between 59th Street and 110th Street. To the west of the Park we have the Upper West Side and to the east we have the Upper East Side. Although there are subtle differences between them, to an outsider Upper East and Upper West are both very similar high-end residential districts, with some retail, entertainment venues and museums. Previously, the neighborhoods declined considerably above 96th Street or so, but these days, it is pretty good until 110th Street or above. Before the 1970s, the Upper West Side was pretty dodgy–“West Side Story” was about gang conflicts in the area–and the Upper East Side was traditionally the “Silk Stocking District.” Now the two sides have largely equalized, though the highest-end residences are somewhat more on the East Side than the West.
Above the Park and 110th Street is, broadly, Harlem, which extends into the 130s. Other than the area around Columbia University (116th Street & Broadway on the West Side), it is largely a minority (black & Hispanic) residential district, with little commerce besides very locally-oriented retail.
Above Harlem, is almost entirely Hispanic, other than small pockets. It is most generally thought of as Washington Heights, though there are many sub-neighborhoods. Like Harlem, it is largely residential, with only locally-oriented retail.