Shingon's central claim is stated in the title of one of Kukai's works: that buddhahood is attained in this very body, in this life, and not after countless lifetimes.
The two realms
The teaching is carried by a pair of mandalas used together. The Womb Realm sets out the world as the compassionate unfolding of the Buddha Mahavairocana, arranged around a central lotus. The Diamond Realm sets out the same reality as indestructible wisdom, in nine assemblies. Neither is complete alone. A practitioner is formally introduced to both at initiation, by throwing a flower onto each to determine which figure the practice will centre on.
The three mysteries
Body, speech and mind are brought into correspondence with the Buddha's own: mudra in the hands, mantra in the mouth, and the visualised presence of the deity in the mind. Performed together, these are held to make the practitioner's activity and the Buddha's activity indistinguishable, which is the mechanism behind the claim about this body. The rites are transmitted from master to disciple and are not published, which is what the word esoteric is doing in the school's name.
A reading of the whole of Buddhism
Kukai arranged all teaching into ten stages of mind, running from the animal state through Confucian ethics, the Hindu and Hinayana positions, and the other Japanese schools, to the esoteric teaching at the tenth. The scheme does not reject the earlier stages; it treats them as accurate accounts of where a mind actually stands, which is why Shingon absorbed rather than displaced what it found in Japan.