A nonlibidoist is a person who does not have a sex drive, and hence does not experience sexual urges or desires (and in particular, does not masturbate). Nonlibidoism is not equivalent to asexuality, since a large percentage of asexuals do have sex drives or libidos, but still lack any sexual attraction.
The term "nonlibidoism" was used by the (now defunct) Official Nonlibidoism Society. Due to the popularity of a more inclusive definition of asexuality, the Official Nonlibidoism Society ceased to use the term "asexual" for its members, believing that it had "by now become almost synonymous for solo-sexual [or] masturbator". Nonlibidoism was a much more stringent definition than AVEN's standard description of asexuality, for people who have no sex drive, and have never had one. Some people considered nonlibidoism the only valid form of asexuality.
Most people on AVEN believe that nonlibidoism is simply one among many valid varieties of asexuality, neither better nor worse than being an asexual with a sex drive.