The three mysteries in correspondence, buddhahood in this very body. It is a different thing from Tibetan tantra.
Tangmi existed in China for a short time. From the arrival of the three great masters of the Kaiyuan era in 716 to the Huichang persecution in 845 is under a hundred and thirty years. What it left behind has lasted a great deal longer: much of the ritual performed in East Asian monasteries today is Tangmi's inheritance.
The yoga flaming-mouth rite, the water-and-land assembly, the Great Compassion Dhāraṇī, the Śūraṅgama mantra, the Cundī mantra. The pieces of Buddhism most familiar to ordinary people all start here. A school that "disappeared" is being recited every day.
The three mysteries in correspondence: the hands form a mudrā, the mouth recites the mantra, the mind visualises the deity. When all three correspond at once with the three mysteries of the Buddha, buddhahood in this very body becomes possible, with no three incalculable aeons in between. This is the sharpest divide between esoteric and exoteric Buddhism.
The dharmakāya teaches: exoteric Buddhism holds that the teaching is delivered by the manifested body of the Buddha (Śākyamuni). Esoteric Buddhism holds that the dharmakāya Mahāvairocana teaches directly. That difference settles how the tradition rates its own standing.
The two maṇḍalas: the Womb Realm (from the Mahāvairocana Sutra, standing for principle, the awakened nature every being already has) and the Diamond Realm (from the Vajraśekhara Sutra, standing for wisdom, the wisdom achieved). The two joined together make the complete system of Tangmi.
The Sutra of Mahāvairocana's Awakening and Miraculous Transformations. The basis of the Womb Realm, and the source of "bodhicitta as cause, great compassion as root, skilful means as culmination".
The basis of the Diamond Realm, and the source of practices such as the five-stage realisation of buddhahood.
General ritual and methods of attainment, weighted especially heavily in Tendai esotericism.
Yixing (683–727) was a Tang astronomer as well as an esoteric monk. His commentary is the most important text for understanding Tangmi, and it brought Chinese cosmology into esoteric interpretation.
The theoretical foundation of Japanese Mikkyō. Kukai is one of the most important figures in the cultural history of Japan.
Esoteric practice requires empowerment and authorisation from a teacher. That is a hard requirement. Practising esoteric methods with no transmission counts inside the tradition as ineffective or worse. It differs completely from exoteric practice, and it is why esoteric material is not published openly.
Systematic training in four stages: the eighteen paths, the Diamond Realm, the Womb Realm and homa. The standard entry sequence in Japanese Mikkyō, usually taking months to years.
Descended from the Vedic fire sacrifice in India, taken over and reworked by Buddhism. It divides into rites for pacifying, increasing, subduing and attracting.
Contemplation of the syllable A is the most basic and most central practice in Japanese Mikkyō: you contemplate the Sanskrit letter A, which stands for the originally unarisen. Relatively speaking, it is one of the few esoteric practices that need no elaborate preliminaries.
A point worth getting right: Tangmi and Japanese Mikkyō are a different thing from Tibetan tantra. The difference is substantive. Tangmi belongs to the middle period of esoteric Buddhism (the caryā and yoga classes) and has no highest yoga tantra. So there is no consort practice, no ritual consort and no tulku system. Tibetan Buddhism includes highest yoga tantra, and both its content and its lines of transmission run differently.
On whether Tangmi was "lost": the accurate statement is that the school form disappeared while a great deal of the ritual survived. In the Republican period Chisong, Wang Hongyuan and others went to Mount Koya in Japan to receive the transmission and bring it back. (Note that what Nenghai and Fazun introduced was Tibetan tantra, not Tangmi.)
A third point: esoteric practice runs on the teacher-student relationship and on authorisation, which makes it the branch most open to abuse in any tradition. If someone claims a transmission and the relationship also involves demands for money, control over your life or sex, walk away.
For most people Tangmi is not a path you can take up on your own. Its structure requires transmission.
As something to know about, Yixing's commentary and Kukai's works are both worth reading. If you only want to see what it looks like, go and look at an image of the Womb Realm and Diamond Realm maṇḍalas. They carry the highest information density in Buddhist art, and every position on them means something doctrinally.