Hi there!
The United States of America's Constitution divided the Government into three different branches, the legislative, executive and judicial. This meant that each branch will be having their specific powers, and it set up something defined as checks and balances. These consisted on make sure no one branch would be able to control too much power, the division of powers to rule the country.
Most of the countries in the U.S. have several equal powers in the same branches. For example, the legislative branch which is in charge of making laws but they must consider the opinion of the President to set up a law too. Some of the U.S. ex-presidents like George Washington and Thomas Jefferson put the veto threats in action.
New York city,one of the most populated cities in the country is different to some of the cities around the country. The head of the executive power is not the President but the governor. The legislature is conformed by the Senate and the Assembly, and the judicial power is ruled by the City's Unified Court System instead of the Supreme Court in other American cities. To be more precise about what exactly are the duties and responsabilities of each branch, I will mention several details about each one of them.
First, the executive power is not only in charge of the governor itself but his departement too. The current elected executive officers are: the governor, whose is the head of the executive branch and commander-in-chief of the state's military and naval forces. The lieutenant governor, also known as the vice-governor, whose role is precise within the branch and its rank is varied by jurisdiction. And the comptroller, whose in charge of auditing government operations and operating the state's retirement system, known as the New York State Common Retirement Fund. These positions are currently ruled by Andrew Cuomo, Kathy Hochul and Thomas DiNapoli.
The legislative power or legislature is the lawmaking branch of state government. It is composed of the Senate and the Assembly. The Lieutenant governor is the Senate's President -currently Kathy Hochul- whose in charge of presiding each legislative sessions and rarely uses casting vote to break ties in measures before the house. The Assembly President is a speaker, who is elected from and by the same Assembly membership. Although other countries consider this power as the most "powerful" one, this has the same weight as the other in New York city.
Finally, the judicial power or branch is presided by the New York State Unified Court System, which is the one that interprets and applies the law of New York, ensures equal justice under law and provides several mechanisms for dispute resolution. The judicial system is composed of the trial courts, consisting of the superior courts, local and appellate courts. The duties of this branch is to keep justice to any citizen who deserves it.