around this time period (11th-12th century) the Roman Catholic Church started getting ideas about Papacy being the embodiment of Christianity, as opposed to a more apostolic approach. This leads to Crusades, generally as a concept, for a big mess of social, theological, and political reasons. Crusades against "Infidels" in the "Holy Land" (~1000 - ~1700), against Pagan and Eastern Orthodox Slavs in the Northern Crusades (~1100 - 1400), against schismatics and heretics (what we're talking about here ~1200 - ~1400), Reconquista of course (~700 - ~1500), and "popular" crusades ("unsanctioned" by the church, basically powerful zealots taking it upon themselves.)

Talking about why various crusades were waged you really gotta look at the specific events tbh.

What heresy were they supposedly committing

So broadly speaking, heterodoxy. Like I said the catholic church got the idea that it embodied and defined Christianity, so deviation from that party line undermines their position as the arbiter of all things spiritual. Obviously, there are going to be alot of people over time with differing theological ideas, so you get many reformist movements over time.

So, specifically speaking of the Cathars they most famous for being anti-clerical (opposition to religious authority) and quasi-dualist. Getting into dualism and gnosticism broadly is something I'm not sure is worth going into in detail here. To be extremely vulgar about it Catharism has two opposing deities. Good God, (New Testament) creator of the spirit, and Evil God, (Old Testament) creator of matter and the physical world. Good and Evil gets worked out as the two forces keep each other in check. Real East meets West stuff. They reject Christ's resurrection and the cross iconography because they believe in reincarnation instead, view Baptism as a false sacrament (going so far as to say John the Baptist was an agent of evil), are big fans of vows of poverty, among many other things.

and what material factors drove the Albigensian Crusade?

So like I said earlier Crusades as a concept have their own driving factors but IRT the Cathars specifically it is largely accepted that it was about pacification of the Languedoc or Occitania in what we think of today as Southern France. Again, this in itself is its own big topic.

The name "Albigensian" comes from the city most associated with the Cathars, Albi. The region was culturally closer to Catalans than French, and the most powerful noble in the area the count of Toulousse was politically closer to the Angevin (Plantagenet) who ruled Aquitaine to the West than the French Capets in the North. Nominally the crusade was about eradicating the Cathars but the regions Catharism flourished in were ruled by minor nobles who embraced the religion (if not actually practicing the Cathar lifestyle, remember those vows of poverty?) as one part of remaining independent from France.

There's a ton of stuff I didn't mention, these are very interesting topics and each worth looking into in their own right.