Vaishnavism's internal divisions are teacher-lineages rather than schisms, and the major ones are named for the philosophers who defined their theology.
Ramanuja, in the eleventh and twelfth centuries, taught vishishtadvaita, qualified non-dualism: the soul and the world are real and are the body of God, distinct from him and wholly dependent on him. That is the Sri Vaishnava tradition of the south, which grew out of the Tamil hymns of the Alvars. Madhva taught outright dualism, the soul permanently distinct from God. Vallabha and Nimbarka produced further positions. Chaitanya, in sixteenth-century Bengal, founded the Gaudiya tradition of ecstatic congregational singing, from which the twentieth-century Hare Krishna movement descends. The Ramanandi order, from the fourteenth century, is by some accounts the largest monastic order in Asia.
Where the tradition came from is genuinely unresolved, and the scholarly accounts compete. One holds that it formed by the merging of several non-Vedic devotional cults (of Vasudeva, of Gopala Krishna, of Narayana), which were then attached to the Vedic Vishnu, with the doctrine of avatars serving as the mechanism that made the merger orthodox. Another traces identifiable stages of development from the last centuries BCE. A third argues that the evidence supports an early southern Krishna devotion, against the usual assumption of a northern origin.
What is documented early and hard is the Heliodorus pillar at Besnagar, raised in the second century BCE by a Greek ambassador who described himself as a devotee of Vasudeva. It places the worship, and a foreign convert to it, well before the texts that describe either.