The case of Texas v. Johnson was decided in 1989 and is a very famous and controversial case. In this case, the Supreme Court held that laws against the burning of the United States flag are unconstitutional.
Basically the Court reasoned that these types of laws are aimed at the content of speech. These laws are saying that a certain message (presumably, in this case, the message that you dislike the flag or what it stands for) may not be expressed in the United States.
The Court held that such a law is a violation of the First Amendment.
The court ruled that burning such a flag, in this case, our American flag was protected by the First Amendment right to protest.