A self-governing body with a censorship code, most visible (and most likely to receive mention) when that code is so draconian and unreasonable in its restrictions that it cripples an entire artistic medium. It usually comes about due to the pleas of the Moral Guardians and Media Watchdogs, but sometimes, it will come about due to some other circumstances (e.g. the beliefs of a specific corporation); regardless, the presence of such a Censorship Bureau will be a bane to the medium and usually result in it not being taken seriously.
Note that most of these were created so that an even more draconian code wouldn't be forced on the medium from outside (although with many of these examples, it's hard to imagine how they could be any worse).
This article from The Other Wiki lists government media rating organizations from around the world.
See also Animation Age Ghetto for a similar phenomenon.
In France, the law about publications intended for youth was really strictly enforced in 50s to 80s. Guns were sometimes replaced by pointed fingers with no explanation. Foreign comics were banned under any pretense (sometimes because censors were convinced these comics were violent or immoral, sometimes just because they didn't want foreign comics) — not just non-Europeans, even Belgian ones, which is funny because most French honestly believe some major Belgian comics to be French. It was so bad that Spirou magazine had some series whose main purpose was to enhance their image for Censorship Bureau — which for example, glared at Buck Danny because the hero was American.
Another effect of this law was that authors were discouraged of showing romance or even female characters to avoid being censored. Thus a lot of occurrences of Celibate Hero or Heterosexual Life Partner, but also tomboyish (at best) female characters in the comics of that time.
After World War II it was prohibited in Finland to do anything that could be seen as "damaging the relationship between Finland and Soviet Union". This led to self-censorship where people who translated foreign comics often changed the words so that they could not be seen as anti-communist. One example included changing "These watches from workers paradise are worthless" to "These McDuck watches are worthless". In another one phrase "Viva la Revolution!" became "Turn off the lights!"
The "Hays Office" was this for American films during the Golden Age of Hollywood. It was established in 1922 as the movie industry's answer to state censorship initiatives, but The Hays Code wasn't much enforced until 1933. Joseph I. Breen succeeded Will H. Hays in 1945. The office was finally dissolved in 1968.
Nowadays, the British Board of Film Classification (BBFC) must approve every film and DVD released in the UK. Items not approved may not be legally sold. This created a controversy in 2007 over the "banning" of the video game Manhunt 2. It should be noted that the BBFC nowadays concentrates on age-rating films, and officially declare that the only thing that will get a film banned outright nowadays is rape and torture filmed in an overtly erotic way.
The American FCC, since it a) charges very large fines for violating the standards for programs in media it has authority over (mainly broadcast television and radio), and b)refuses to publish a list of those standards; they can't file criminal charges. Those fines started at $27,500 and went up to $325,000 after Janet Jackson flashed the audience at the 2004 Super Bowl. The record is $1.2 million, for a FOX reality show called Married by America in which someone licked whipped cream off a woman's censored nipples.
Apple's iOS App Store has attracted controversy for censoring certain apps, including those from fashion magazines, papers with PageThreeStunnas, and those with politically controversial material, such as one that sent alerts about US Government issued drone strikes.
In 1986, Nintendo of America — which saw video games as family entertainment — leaned on third-party publishers to remove all references to religion, drugs, alcohol, and adult situations from their games. Nintendo threatened to withhold the "Nintendo Seal of Quality" and not manufacture the patented game cartridges for companies that didn't comply. Even when Sega made a name for itself with the Genesis, Nintendo controlled enough of the video game market share to profoundly influence how games were localized for western consumption; however, Sega's marketing was successful in giving Nintendo a "kiddie" label that it's been cursed with ever since, even after their policies (mostly) lapsed with the advent of the ESRB, which established an independent rating system allowing games to be targeted appropriately.
Keep in mind that these were the very conservative Eighties during which the US Congress got extremely close to censoring music. If they could censor an art form Older Than Dirt, there's no telling what they would do to New Media like video games that hadn't yet (and still hasn't) gained respect as True Art. Nintendo likely feared that the fledgling industry could be shot down by censor-happy Moral Guardians if they weren't careful. In fact, these fears were almost realized in the Nineties when congressional hearings were held over the violence in Mortal Kombat.
Sega themselves censored some of the games they released, most notably Streets of Rage III; they also instituted an MPAA-style rating system on many of their games before the ESRB was formed.
The irony is, the "Seal of Quality" was meaningful then — Nintendo also blocked truckloads of crap games. After the courts ruled that Nintendo could no longer prohibit independent games, scads came out without the seal — but stores started voluntarily refusing to carry unlicensed games due to their bad reputation. Nintendo did apply pressure to many retailers, famously threatening to withhold licensed games and consoles from any retailer that carried unlicensed games, but even those retailers brave or foolish enough to stand up to Nintendo's tactics soon stopped selling unlicensed games after being flooded with returns of games that didn't work or were nigh-unplayable.
England had a Censorship Bureau of sorts for theatre which lasted from the Elizabethan era until the late twentieth century. Some believe that operettas/musical theatre came about, at least in part, as a way of getting around this, since musical performances didn't fall under the Censorship Bureau's jurisdiction. This duty was held by the Lord Chamberlain's office; whilst the title and office still exist, it hasn't been responsible for censorship since 1968 (the Lord Chamberlain is today in charge of coordinating the Royal Household and communicating between the House of Lords and the Sovereign; he and his office are basically the Queen's event planners).
One way around these restrictions was to declare performance venues to be members-only clubs, which were immune to government oversight. Prospective viewers would pay a nominal membership fee (often included in the price of a ticket) and go in. While this had obvious benefits for proprietors of strip clubs and the like, it also enabled the performance of more edgy plays like Look Back in Anger which, despite their quality, could not be staged publicly.
Dictatorships always had their Censorship Bureaus, more concerned with political references than sex and violence.
A rather fascinating case of this was the banning of Voltes V in the Philippines due to the central anti-dictatorship conflict, which actually lead to the Super Robot being seen as a symbol of revolution.
Another interesting case was the release of The Who's Quadrophenia in 1973 in Spain. They censored the liner notes for objectionable lyrics, and deleted "Doctor Jimmy." Ah, but that's not all. The booklet contained pictures illustrating Jimmy's story. On the wall of Jimmy's room, there were shall we say, clothing deficient women. These booklets were printed outside the country, so the censors had each offending picture properly clothed with a Black Bra and Panties...by hand.
Caesar surprisingly averted this; Marcus Porcius Cato had been one of his most implacable enemies during the civil war and the two disagreed (sometimes violently) on most political matters, yet the only action he took when seeing literature written after the war and during his dictatorship praising Cato was to write his own piece pointing out his (Cato's) faults. It was acceptable to criticise Caesar during his dictatorship. (You might even make it to important offices, like in the case of Marcus Junius Brutus, who had fought on Pompey's side and written a treatise on Cato after that, praising his virtues.) The emperors after him did not follow this policy and played the trope relatively straight.
ALL governments engage in censorship under sufficient pressure, such as total war. Go to That Other Wiki and enter "Office of Censorship" for a prime example.