When parts of the image (typically graphic or sensitive contents such as sexual organs, gore, or offensive symbolism) were obscured, typically to comply with censorship laws.
On Danbooru, this typically encompasses various Japanese pornographic materials that were censored as a result of the Japanese law. Techniques that prevent viewing sexual parts, usually vaginas, penises and dildos but occasionally nipples. Note that the anus is not usually considered a sexual part in Japanese censorship.
Common forms include mosaics, black bars, mysterious white lights, inverted colors, shaded bars, and the infamous invisible penis. Tentacles or phallic objects are commonly used to resemble genitals and generally don't need to be censored, though the point of intercourse with a vagina does.
For obstructions that are internal to the picture, like a lock of hair that just happens to cover the genitals, use the tag convenient censoring instead of censored.
History
Obscenity in Japan has been illegal to some degree since the 19th century Meiji Era, but the Criminal Code of 1907 relating to obscenity was reinforced (or at least not overturned as other laws restricting freedom were) when American General MacArthur established Japan's modern laws after WW2.
Criminal Code of Japan, Article 175. (Distribution of Obscene Objects)
A person who distributes, sells or displays in public an obscene document, drawing or other objects shall be punished by imprisonment with work for not more than 2 years, a fine of not more than 2,500,000 yen or a petty fine. The same shall apply to a person who possesses the same for the purpose of sale.
Contrary to popular beliefs, Article 175 does not enforce the censorship of pornographic materials. Rather, it is the publisher of these pornographic material who "self-censored" their work as a form of loophole around Article 175, as this technically transformed the work into an "erotic" material by not fully showing the genitalia, which results into less restrictions from the law. This practice was first done by the Japanese pornographic aka the "Adult Video" (AV) industry in 1970s and has spread to other mediums.
Note that the law relates to distribution within Japan; it is not illegal for Japan-made porn intended only for export to be uncensored.
Obscenity laws are gradually being relaxed, but not eliminated. Pubic hair is no longer required to be censored, for example. There are now attempts to repeal the Meiji-era/post-WW2 censorship law, though this has yet to reach deliberation.
