What are the names of some organizations that support censorship?

Dictatorships and other authoritarian regimes are usually strong supporters of censorship, seeking to keep any information that may threaten their rule out of the hands of the people. However, I assume your question is about pro-censorship groups in otherwise free societies.

Generally, groups that support censorship of media are conservative in nature, seeking to either protect a notion of traditional family values or religious values.

In the United States, most pro-censorship groups are Christian in character, but some bill themselves as parents' groups, in the mold of the Parents' Music Resource Council of the 1980s, a group founded by several Senators' wives. The PRMC was successful in convincing most music labels and retailers to adopt the "Parental Advisory" labels seen on CDs with coarse language.

Following the Sandy Hook Elementary School shooting last year, the renewed debate over gun violence has sparked something of a renaissance in pro-censorship groups who argue that media violence causes real-life violence. The merits of this argument are for another article, but know that this is a common justification cited for censorship.