How does the Judicial Branch check the power of the President?

The Article III federal courts (ultimately, the US Supreme Court) uses judicial review to declare Presidential Executive Orders invalid if they are in conflict with the Constitution or if the Executive Order breaches the separation of powers (for example, involves an action, like declaration of war, which is delegated to Congress). The Courts are the interpreters of the laws and as such they interpret the Constitution and laws to decide if they conflict with one another.

The US Supreme Court may only declare an Executive Order unconstitutional if it is relevant to a case or controversy before the Court. They cannot not take preemptive action against the Executive Branch (or the Legislative Branch).

Chief Justice John Marshall clearly affirmed the power of judicial review in the case Marbury v. Madison, (1803), when the Court declared Section 13 of the Judicial Act of 1789 unconstitutional.