1) Chalcedonians are termed after the Council of Chalcedon that was convened in 451 to condemn the Monophysite heresy. It basically reaffirmed the paradoxical non sequiturs of the unitary dual nature of Christ and of tritheist monotheism that have become orthodox Christianity (which includes both Roman Catholicism and Eastern Orthodoxy).
2) Manichaeism was a dualist religious creed perhaps best described as falling in that class of religions (such as Daoism and Zoroastrianism) that view the divine realm as equally balanced between the force of good and the force of evil (though they arrive at this by varying rationales). The founder, Mani, regarded himself as a Christian, but Manichaeism is so far removed from the what became the 'mainstream' of Christian theology that it is almost always referred to as if it's a separate religion, rather than a Christian heresy. That may be accurate, since it was essentially a synthesis of Christian, Persian, and Buddhist concepts.
3) A key distinction of Manichaeans is that most of them embraced the concept of the phenomenal world (Earth and humanity) as an evil realm. This was a very Gnostic concept, although Christians in general tend to agree, just more ambivalently. It's also a very Buddhist concept, and perhaps foremost a Buddhist concept because some scholars argue that Basilides and Valentinus (who were the founders of the most 'recognizably Gnostic' creeds) were influenced by Buddhist thought. However, it's worth noting that Manichaeism is thought to have more in common with Marcionism than with Valentinism. Marcion's heresy was the (very reasonable) notion that the Old Testament god was vindictively evil and obviously different than the benevolently good God described by Christ. Anyhow, the most extreme Manichaeans even considered childbirth (and procreative sex) evil because it perpetuated the alleged evil of the material realm.
4) The great opponent of the Manichees was St Augustine, who had himself been a Manichee hedonist in his youth, but conveniently saw the light once he had his fill of debauchery (see The City of God). After St. Augustine, Manichaean were revived amongst the Paulicians of Anatolia and the Priscillianists of Iberia. Priscillian was basically a hard-core mystical ascetic (for which he became the first of many, many Christians to be executed for heresy). By contrast, the Paulicians rejected the Church hierarchy and the orthopraxy of Christian conduct.
Historically, the very same two responses always emerge from any creed (such as Gnosticism, Buddhism, or Manichaeism) that regards the phenomenal world as evil: (a) the encratite response holds that salvation is to be found by strict asceticism; that since the world and human nature is evil, one must refrain from behaving human; (b) the antinomian response holds that since the phenomenal world (and especially the human body) is evil, that it doesn't matter what one does with one's body, because only the soul matters. Oftentimes, the idea is that sensory experiences of whatever kind aid the soul in achieving enlightenment and transcendence.
They are just two extremes of the weird tendency of humanity to abhor and debase itself.
5) Getting back to the point, the Albigensian Catharists of Aquitaine Spain were the heirs of the Priscillian heresy, and therefore indirectly Manichaean (and therefore indirectly Marcionite). As for the Paulician heresy, its last gasp was amongst the Bogomil heresy of the Slavs (which to be exact was a synthesis of Armenian and Slavonic gnosticism). In any event, the core tenet of the Albigensian creed was that the material world is evil, created by the "demiurge" god of the Old Testament ("demiurge" means the 'fashioner' or 'designer' of the universe, since only an evil, fallen god would fashion such an evil, fallen realm). Albigensians believed in reincarnation and that the way to escape the cycle of rebirth was by absolute asceticism, which is why they were called Catharists (which means 'cleansers'). Although, bear in mind that their moral system was upside down by more familiar standards (they had less of a problem with homosex, by example, since it did not result in childbirth, than with procreative sex, especially nonmarital fornication).
So, to explain my comment before, the Albigensians were branded as evil heretics and (since they were also very wealthy) Pope Innocent III invoked a Crusade and then an Inquisition against them in the 13th century and they were obliterated.
I hope that helps clear things up!
Correction: Aquitaine France, not Spain.
My knowledge of all of this is close to the null set. Thanks for taking the time with me. It is interesting how worked folks get over how many angels dance on the head of a pin. But then, the real game, as always, among the leaders, is power. Splinter groups tend to disrupt, and thus must be contained or eliminated as convenient, or necessary. These days, religion is not really the road to power, except in the Muslim world. That I suspect in part is why Muslims provoke such angst. They remind us of where we once were, a place that now we view with confusion, fascination, and horror.