Manhattan
Brooklyn
Queens
Bronx
Staten Island (though it should probably be part of jersey)
No. Not a term for neighborhood. Each borough has countless neighborhoods.
Brooklyn: Flatbush, Dumbo, Redhook, Brooklyn Heights, Bed-Stuy, Cypress Hills, Cobble Hill, Carrol Gardens, ....
Manhattan: Soho, Noho, E. Village, W. Village, Upper E. Side, Murray Hill, Theater District, Upper W. Side, Chelsea, Flatiron district, Tribeca, Harlem, Hells Kitchen...
There you have it.
I believe Brooklyn was actually it's own city and not part of NYC until pretty late ? maybe the 40's ? I can't remember exactly. Had their own baseball team (Dodgers.)
Wierd thing about living in Brooklyn (to me at least) were the people I met who lived there their whole lives and only went to Manhattan once a year or less, even though it was 15 minutes away by train.