Censorship is the prohibition of speech from a medium or audience for ideological reasons. In another meaning, it is the restriction of a speech act for a content-specific reason.
What is Not Censorship
There is a tendency of individuals with particularized political beliefs to conflate "censorship" with "opposition," especially when a desired effect in the public sphere is stymied. However, one must remember that a political goal may be opposed for content-neutral reasons, making its opposition not censorship, but disagreement with cause. For example, someone arguing against the teaching of holocaust denial in public schools would not be censoring holocaust denial, but would be opposing the teaching of a malicious untruth.
The First Amendment and censorship in the U. S.
In the United States, the First Amendment states that "Congress shall make no law... abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances." Broadly speaking, the First Amendment is designed to prevent the government from exercising censorship. However, the government sometimes censors political and religious speech anyway.
More specifically, the government should not exercise "prior restraint." That is, a citizen should not need advance permission from the government in order to publish something, unless it threatens national security. This does not mean that publication may not have consequences: a citizen can be sued for publishing libel, or incarcerated for disclosing military secrets, but the consequences typically occur after publication, not before.
Censorship is sometimes applied to prohibit obscenity that goes against common standards of public morality; however, these standards tend to fluctuate depending on time and place. Certain language and images that may have been censored in the past are typically common fare in the American media today. On the other hand, while nudity, for example, may be acceptable on mainstream French television, that is much less likely to be the case in conservative Muslim countries.
The word censorship derives from the name of the Roman officials who conducted the census and supervised public morality. Cato the Elder was a famous censor, who during his tenure sought to ensure that Romans did not exploit their opulence to the detriment of the city and its people. Unpopular in his time, Cato became symbolic of the Roman ideal in later years.