The Tathagatagarbha sutras, appearing in India from around the third century, teach that all sentient beings contain the womb or embryo of a Buddha. Their characteristic mode is imagery rather than argument: gold buried in filth, a lamp inside a covered jar, a treasure under a poor man's house, honey guarded by bees, a royal embryo carried by a destitute woman. In each the valuable thing is already fully present and merely obscured, and awakening is uncovering rather than construction.
The doctrine answered a real question. If beings are empty and driven by conditions, what accounts for the possibility of awakening at all, and why should anyone believe it is available to them? Buddha-nature supplies a positive answer where emptiness gives a negative one, and it did so in a register accessible to ordinary practitioners.
In East Asia it became foundational rather than supplementary. The Chinese reception, mediated by the influential treatise known as the Awakening of Faith, made Buddha-nature the ground of Chan, Tiantai, Huayan and Pure Land alike. Chan's conviction that awakening can be sudden follows directly: if the nature is already present, no gradual accumulation is strictly required to produce it. Pure Land's assurance that the incapable are not excluded rests on the same base. Nichiren's teaching that Buddhahood is attainable in this very body draws on the Lotus Sutra's related claim of universal potential. Chinese and Japanese thinkers extended the doctrine further than Indian sources had, to grass, trees and the land itself.
Zen's best-known koan turns on it: asked whether a dog has Buddha-nature, Zhaozhou answers wu, no, in flat contradiction of the doctrine every questioner knew, and the contradiction is the point.