Short answers, pulled from the story.
The Tathāgatagarbha sūtras are a group of Mahayana Buddhist texts presenting the concept that every sentient being carries within them the womb or embryo of the Buddha, called the tathāgatagarbha. They teach that Buddhahood is already present in all beings and requires uncovering rather than cultivation. Key texts include the Tathāgatagarbha Sūtra (c. 200-250 CE), the Śrīmālādevī Siṃhanāda Sūtra, and the Mahāyāna Mahāparinirvāṇa Sūtra.
Tathāgatagarbha is a Sanskrit compound combining "tathāgata," meaning "the one thus gone" (a title for the Buddha), and "garbha," meaning root, embryo, or essence. Together the term refers to the Buddha-embryo or womb of Buddhahood said to reside within every sentient being.
Scholars including Sree Padma, Anthony Barber, and Brian Edward Brown associate the inception of the Tathāgatagarbha doctrine with the Mahāsāṃghikas of the Āndhra region in India. Brian Edward Brown dates the Śrīmālādevī Siṃhanāda Sūtra to the Andhra Ikshvaku period in the 3rd century CE as a product of this community.
The Laṅkāvatāra Sūtra, dating to the 3rd century CE, directly addresses this question in Section XXVIII, where the disciple Mahāmati asks if the tathāgatagarbha is the same as the ego-substance of Hindu philosophy. The Buddha's response insists they are not identical: the tathāgatagarbha is taught as a means to help listeners overcome fear of egolessness, and is said to be identical to emptiness. The same text's concluding Sagathakam section, however, does not deny the reality of what it calls the "pure self."
The Ratnagotravibhāga, also known as the Uttaratantra or "The Supreme Continuum," is the only Indian treatise attempting to build a coherent philosophical model from the Tathāgatagarbha sūtras. It draws especially on the Śrīmālādevī Siṃhanāda Sūtra and became very influential in Tibetan Buddhism. Despite the appeal of tathāgatagarbha ideas in East Asian Buddhism, the Ratnagotravibhāga played a relatively small role there because East Asian Buddhism prioritized sutra study and the Awakening of Faith.
Paul Williams notes that the Tathāgatagarbha sūtras were comparatively less prominent in India than the Yogachara and Madhyamaka schools, but became increasingly significant in Central Asian and East Asian Buddhism. Key texts including the Mahāyāna Mahāparinirvāṇa Sūtra were substantially extended in China, and the Awakening of Faith was composed in China rather than India, making East Asia the site of the doctrine's fullest development.