"Prior restraint" means that the government can prevent or prohibit speech (or publication of something before it can be published) only if the content can be shown to present a clear and identifiable danger to public safety.
For the government to take such measures--blocking a form of speech or communication before it is allowed to happen--the form of speech must be such that it would clearly pose a threat of extreme danger to the community -- or might cause severe risks to national security. An example of the second sort of "prior restraint" claim happened in 1971, when the US government tried to block the New York Times and the Washington Post from publishing classified documents which were known as "The Pentagon Papers," regarding government handling of the Vietnam War. In that case, the Supreme Court decided, in New York Times Company v. United States, that the government's concern for security did not override the 1st Amendment rights of the news media to publish the information which they had received. The Court did not agree that the release of the government information would cause inevitable, specific, immediate danger to the United States. So you can see, proving the need for "prior restraint" is a very high bar to get across, and 1st Amendment rights usually prevail.
