Immigrants arriving by ship in the port of New York City between 1855 and 1954 would have processed through one of two stations: Castle Garden (state processing center from 1855 - 1890) or Ellis Island (federal processing center from 1892 - 1954).
Naturalization is the voluntary process by which an alien becomes an American citizen. Naturalization could take place in any court of record, and for many immigrants that meant the court most convenient to their residence regardless of the name or type of court, which varies from state to state. Generally, naturalization was a 2-step process requiring 1st the declaration of intent, followed by the petition for naturalization. The two steps of the process were not required to take place in the same court.
For a more in-depth introduction to naturalization records, see Naturalization Records, by the National Archives, particularly the first three sections: Introduction, The Courts, The Process.