Short answers, pulled from the story.
The earliest seeds of Advaita Vedanta took root in the Sannyasa Upanishads during the first centuries CE. These writings predate the famous philosopher Adi Shankara by several hundred years and established a distinct non-dual outlook within Hindu thought.
Gaudapada wrote the Mandukya-karika in the 7th century. He adapted philosophical concepts from Buddhism to give them a Vedantic basis and interpretation while introducing ideas about the illusory nature of the world that would later become central to the tradition.
The term Advaita literally means not-two or one without a second. Only Brahman, the one, is ultimately real while prapanca, the world or multiplicity of thought-constructs, is not fully real.
Adi Shankara proposed three levels of reality to explain how the empirical world relates to absolute truth. Paramartha represents the state of experiencing what is absolutely real, Vyavahara consists of the empirical or pragmatical reality, and Pratibhasika describes apparent reality based on imagination alone.
The tradition gained institutional power with the ascent of the Sringeri matha and its jagadguru Vidyaranya in the 14th century within the Vijayanagara Empire. Before this period, no mention of these monasteries can be found before the 14th century CE according to Paul Hacker.