A very useful historical overview of Courts in NYC can be found in the Encyclopedia of New York City. (pp. 321-325) Reference - F128.3 .E75 2010 available to CUNY users as an e-book,
"Inferior Court." West's Encyclopedia of American Law. Ed. Shirelle Phelps and Jeffrey Lehman. 2nd ed. Vol. 5. Detroit: Gale, 2005. 396. Gale Virtual Reference Library. Web. 20 June 2016.
This term may denote any court subordinate to the chief appellate tribunal in the particular judicial system (e.g., trial court); but it is also commonly used as the designation of a court of special, limited, or statutory jurisdiction, whose record must show the existence and attaching of jurisdiction in any given case.
There were several different courts in operation in New York City, under several different jurisdictions. If you are looking for trial documents, including transcripts, it is essential to know in which exact court the trial you are researching took place. Some resources for finding this information include The New York Times and other New York City newspapers which regularly listed and reported upon court proceedings.
General Information on Historical NYC Courts.
Cobb, W. Bruce. (1925) Inferior criminal courts act of the city of New York, Freely available through Hathi Trust. Book available by appointment in our Special Collections Room.
On the reorganization of NY State Courts see: Report and draft of proposed legislation to implement the judicial administration provisions of the new Judiciary article of the Constitution, article VI, adopted by the people on November 7, 1961. Stacks - KFN 5010.62 .C68 1962
New York State (1904) The court rules of the State of New York: including the general rules of practice, the Court of Appeals rules, rules of the Court of Claims, of the appellate divisions of the different departments, special rules of practice in the first judicial department, rules of the City Court and Municipal Court of the city of New York, rules of practice of the Surrogates' Courts of the counties of New York and Kings, and rules of the Court of Special Sessions for the city and county of New York : with notes, references and indexes. Available through Hein OnLine (John Jay login required)
Charity Organization Society of the City of New York. Committee on Criminal Courts. (1918). The forgotten army: six years' work of the Committee on Criminal Courts of the Charity Organization Society of the City of New York, 1911-1917, a story of its work for the clean, intelligent and kindly administration of our inferior criminal courts. [New York: Committee on Criminal Courts of the Charity Organization Society of the City of New York.] freely available on Hathi-Trust
Scott, Kenneth (1983) New York City court records, 1684-1760: genealogical data from the Court of Quarter Sessions. not at John Jay College - Available at Lehman Stacks - KFN5971 .N4 S38 1983 [and many other libraries]
Underwood, Richard H. (2017) Gaslight lawyers: criminal trials & exploits in gilded age New York. Stacks - KF355.N4 U53 2017
Worthington, George E (1925) Specialized courts dealing with sex delinquency; a study of the procedure in Chicago, Boston, Philadelphia and New York, Stacks - KF9448 .W6