What is the main job of the Judicial Branch?

If we think of the "judiciary branch" as referring to all the local, state, and federal courts in the United States, then its job is manifold. It is responsible for, among other things, deciding if there has been a violation of criminal law or civil code and deciding the necessary punishment for any violation.
If we think of the "judiciary branch" as mostly referring to the Supreme Court of the United States, then its function is to interpret the constitutionality of the laws and to serve as a check on the power of the other two branches of government.
However, that particular function of the judiciary branch was a point of contention between Federalists and Anti-Federalists. The sides strongly disagreed about whether or not the Supreme Court should have any right to decide whether or not laws were constitutional.
The Anti-Federalist Brutus argued that doctrine of judicial review would upset the balance of powers:

The power of this court is in many cases superior to that of the legislature. I have showed, in a former paper, that this court will be authorised to decide upon the meaning of the constitution; and that, not only according to the natural and obvious meaning of the words, but also according to the spirit and intention of it. In the exercise of this power they will not be subordinate to, but above the legislature.

The Federalist position--which is the position that eventually won out--was that the Supreme Court needed to have this power. In Federalist 78, Alexander Hamilton argued that the judiciary was the weakest branch because it had no ability to enforce its desires. Thus, it would not be inviting tyranny to grant the Court the power of judicial review:

It is far more rational to suppose, that the courts were designed to be an intermediate body between the people and the legislature, in order, among other things, to keep the latter within the limits assigned to their authority. The interpretation of the laws is the proper and peculiar province of the courts. A constitution is, in fact, and must be regarded by the judges, as a fundamental law. It therefore belongs to them to ascertain its meaning, as well as the meaning of any particular act proceeding from the legislative body.

It is important to note that, despite Hamilton's arguments, Article III of the U.S. Constitution did not specifically grant the Supreme Court the power of judicial review.
However, judicial review was institutionalized very early in the nation's history. Chief Justice John Marshall's famous opinion in Marbury v. Madison (1803) established judicial review in the United States and confirmed that the judicial branch's role is to interpret the law and protect the people from unconstitutional acts by the executive or legislative branches.
https://teachingamericanhistory.org/library/document/the-problem-of-judicial-review/

https://www.law.cornell.edu/constitution/articleiii