What are the SSPX? I've been seeing the conversation around them but am unfamiliar with any of their beliefs or anything (tho them being called "modernist" definitely makes me think their heretics ofc)
It's a truth universally acknowledged that at every major council, a group splits off. At Nicaea it was the Arians, at Trent it was the Protestants, at Vatican I it was the Old Catholics, and at Vatican II it was the SSPX.
SSPX stands for Society of St Pius X, and they basically rejected a lot of the ecumenical and liturgical reforms, and then consecrated a bunch of priests and bishops in direct defiance of the Holy See. Lots of excommunications all round, bad time for everyone. Then a group split off from them for not being traddy enough, and they called themselves the SSPV (Society of St Pius V). Not wholly sure what they're doing now, but they're still around. The Palmarians (super weird cult with their own pope) are also an offshoot of this "Lefebvrist" movement (Archbishop Marcel Lefebvre being the founder of the SSPX).
Currently they're in a "canonically irregular" situation, where they officially denounce sedevacantism and call themselves Catholic, but in practice they don't submit to the Pope's authority, they don't accept the Ordinary Form of the Mass, and they have an entirely parallel structure of priests, bishops, and dioceses. Their sacraments are valid, but illicit (mostly-- they can do confession, baptism and marriage). Basically JPII, Benedict XVI, and Francis have made various moves to get them back in by lifting excommunications, changing rules around the EF, and allowing certain sacraments to be celebrated, but they're still firmly one foot out the door and refusing to budge, which for orthodox Catholics is already too many feet.
SSPX apologists will make all sorts of arguments to say that their rebelliousness is justified and that they're not in schism, but the long and the short of it is that a) every heretic group thinks they're in the right and b) the attitude of non-obedience is really spiritually dangerous anyway.
The "modernist" label is probably better explained by the great labeller, @paula-of-christ, but it's basically in reference to the fact that they think the Church ought to conform to their beliefs, rather than conforming their beliefs to the Church's teachings. It doesn't really matter whether your beliefs are uber traddy and your chasubles are really shiny-- if you're out of step with Rome, you are the problem.
