Party A owns a factory that emits large quantities of foul-smelling
smoke. People in a nearby housing development are annoyed by the odors
whenever the winds shift to a certain direction. Party B, a resident
in the housing development, has lodged a complaint against A for
public nuisance. In what kind of court would the case be likely to be
tried in new york? Why? What would be the probable result of the legal
action?

[Please note that I am not a lawyer and cannot dispense legal advice.]

Here's my belief:

In New York State, the Supreme Courts are courts of general
jurisdiction.  That means that Supreme Court judges hear cases on all
sorts of matters (e.g., breach of contract, personal injury claims,
criminal cases, etc.).  There are Supreme Courts in every county. 
Public nuisance claims may be brought in Supreme Courts.

(Remember that in New York, the "Supreme Court" is a low-level court,
not the state's highest court.  The highest court in New York is
called the "Court of Appeals.")

Without further information, I can only speculate about the exact
court Party B needs to visit for relief.  But it's likely that Party B
should file suit in the Supreme Court in the county in which the
factory and housing development sit.

For an example of a New York case in which a Supreme Court heard a
public nuisance complaint about a factory filed by nearby apartment
dwellers, see:

Pelletier v. Transit-Mix Concrete Corp., 174 N.Y.S.2d 794 (N.Y. Sup.
Ct. 1958).

Another, from the Supreme Court of New York County (which concerns a
gay bathhouse that New York wanted to close in an effort to prevent
the spread of AIDS) is online at:

http://biotech.law.lsu.edu/cases/STDs/St_marks_I.htm

The citation is:

City of New York v. New St. Mark's Baths, 130 Misc. 2d 911, 497
N.Y.S.2d 979 (1986).

(In case you're interested, the Supreme Court granted the City's
request for a temporary injunction closing the bathhouse, and the
Appellate Division handed down a permanent injunction.)

Also, Party B should consider whether the case is best described as a
"public nuisance" or a plain-old private nuisance case.

There's a difference between the two.