From 14th Street south to Houston Street, is the area known as “The Village.” Right around 5th Avenue and Washington Square Park is NYU university, this area, being neither east nor west, is the part most people call ‘Greenwich Village’ (that’s “gren-ich,” BTW)
Between 14th & Houston, East of Broadway over to 1st Ave you’d call it the East Village. East of First Avenue (where the streets are alphabetized, Avenue A, Avenue B, etc.) is sometimes called Alphabet City. It’s a fairly sketchy area.
North of 14th street on the east side in Stuyvesant Town/Peter Cooper Village which is an enormous middle-income housing project that extends all the way up to… 23rd street at least. Maybe 26th, I can’t recall. People wait YEARS on the waiting list to get a Stuyvesant Town apartment. Not that’s its fancy, just safe and (relatively) affordable.
South of Houston on the east side is The Lower East Side, once a Jewish immigrant tenement neighborhood but now an up-and-coming area for hipsters and artists. Certain elements of the old Jewish culture remain (bargain clothing stores on Orchard St., Yonah Schimmel Knishes, and Russ & Daughters smoked fish emporium spring to mind)
The Lower East Side merges with Chinatown somewhat north of Grand Street.
Little Italy barely exists anymore, swallowed up by Chinatown. Just a cluster of restaurants around Mulberry st.
Okay, south of 14th to the West of 6th Avenue is the West Village. It’s the “gay” neighborhood if you will and is a very NICE part of town. Mostly 4 story brownstone type houses, treelined streets.
Whew, I’m all neighborhooded out.