The most expansive vision of the lot: one is all, all is one.
Huayan has exactly one thing to say and that one thing is enormous: any single thing contains, in full, every other thing. Mutual containment in the literal sense, rather than "closely connected".
When Fazang explained this to Wu Zetian he used two famous demonstrations. One was the golden lion in front of the hall (the Essay on the Golden Lion). The other was ten mirrors arranged in a ring with a lamp lit in the middle, so that she could see the endless reflections herself. It is the best piece of teaching in the history of Chinese Buddhism.
Dependent origination of the dharmadhātu: the whole dharmadhātu is one interdependent, mutually containing totality. Any single dharma stands only in dependence on the whole, and the whole shows up complete within any single dharma.
The four dharmadhātus: the dharmadhātu of phenomena, of principle, of principle and phenomena unobstructed (ground and appearance do not get in each other's way), and of phenomena unobstructed among themselves (any appearance contains any other without obstruction). The last of these is Huayan's alone, and the basis of its claim to rank above the other schools.
The ten mysterious gates and the six characteristics: ten sets of relations and six categories (whole and part, sameness and difference, formation and disintegration) used to argue rigorously for how unobstructed interpenetration is possible. This is an argument with a derivation behind it, not a poetic description.
Two main translations, the sixty-fascicle and the eighty-fascicle. Enormous. The "Entry into the Dharmadhātu" chapter (Sudhana's fifty-three visits) and the "Practices and Vows of Samantabhadra" can be read on their own.
The record of Fazang's lesson for Wu Zetian, running the entire theory off one golden lion. Very short, and the best entry text Huayan has.
Fazang's ranking system: the lesser, initial, final, sudden and perfect teachings.
A commentary of vast size, the accumulated body of Huayan scholarship.
One mind with two aspects, suchness and arising-and-ceasing. The theoretical support Huayan and Chan (Zen) hold in common. Fazang wrote a commentary on it.
The ten great vows. The part of the Avataṃsaka most often taken up separately, and a standard item in East Asian daily liturgy.
Contemplation of true emptiness (breaking attachment, seeing emptiness), of principle and phenomena unobstructed (essence and function not two), and of universal pervasion and containment (phenomena containing one another). Set out in Dushun's Discernments of the Dharmadhātu, this is the core of Huayan practice.
Paying respect to all buddhas, praising the Tathāgata, making offerings widely, and on through learning from the Buddha, according with beings, dedicating everything. Huayan takes an unlimited vision and turns it into ten things you can do every day.
Monastic practice forms. The chanted Huayan syllables are one of the most distinctive strands in East Asian Buddhist chant.
"Phenomena unobstructed among themselves" rests on dependent origination, emptiness and the tathāgatagarbha, and is argued strictly, with the ten mysteries and six characteristics as a doctrinal structure you can check. Reading it as lyric feeling is like reading a mathematical proof as an essay.
On why it faded, the plain reasons are these: it is difficult, it is huge, and it never produced a system of practice with a low enough barrier to entry. Its worldview ended up absorbed by Chan, where it became the theoretical floor.
Start with the Essay on the Golden Lion, which is very short. If you still want more, read the Practices and Vows of Samantabhadra, the one part of Huayan you can pick up and use daily.
Worth adding: if you find the Chan lines about "a world in a single flower" beautiful but cannot see what they rest on, they rest on Huayan. Chan's poetry is to a large extent Huayan philosophy after digestion.