NeoCatholicism/Catholicism is a religious denomination.
In the TV series, not much is known about the religion, but given its name, it is obviously derived from present day Catholicism. Whether or not it is controlled/led by the Papacy in Rome is unknown, but it does seem to follow a present-day Catholic hierarchy (such as having Priests and an Archdiocese). It is opposed to the re-sleeving of individuals into new bodies because it believes this puts the soul at risk of hell. NeoCatholics and many other religious groups apply religious coding to their Cortical Stack which prevents them from being spun up or re-sleeved after sleeve death. Citizens born into this community usually have religious coding applied by their parents as soon as their stack is implanted as a one year old but when they reach legal adulthood they can choose to renounce this coding.
In the books
In the books, Catholicism is confined to earth, as Takeshi was unaware of what the word 'Catholic' even referred to. In the books, it is not just re-sleeving after death that the Catholic Church is opposed to, but to all forms of technology that removes the human consensus from the body- needlecasting to other planets, use of Synth bodies, the use of clones, re-sleeving from living body to living body etc. In the books, Catholics seem to be the butt of derision by most people and are targets for killers as there is no fear that they will be brought back to testify. It is stated that the Vatican City has funded an interstellar ship to expand to other galaxies.