Short answers, pulled from the story.
The Middle Way is a core Buddhist teaching with two main aspects: a practical path that avoids both extreme asceticism and sensual indulgence, and a philosophical approach that avoids both eternalism and annihilationism. The practical Middle Way is identified with the Noble Eightfold Path, while the philosophical Middle Way is grounded in the doctrine of dependent origination.
In the Dhammacakkappavattana Sutta, traditionally regarded as the Buddha's first teaching after his awakening, he described two extremes to be avoided: addiction to sensual pleasures, called low and unprofitable, and addiction to self-mortification, called painful and unprofitable. He declared that the Noble Eightfold Path constitutes the Middle Way between these two extremes, leading to calm, insight, enlightenment, and Nibbana.
Eternalism holds that there is an indestructible and eternal self, or that the world is sustained by a permanent metaphysical entity. Annihilationism holds that a person is utterly destroyed at death with nothing surviving. The Buddha's Middle Way, expressed through dependent origination, avoids both by teaching that existence is a current of conditioned phenomena without a permanent self, yet continuing across births through causal conditions.
Nagarjuna was the great Indian master who founded the Madhyamaka school of Buddhist philosophy, whose name means Middle Way. His Mulamadhyamakakarika, known as The Fundamental Wisdom of the Middle Way, directly references the Kaccānagotta Sutta and argues that the concepts of existence and non-existence are both incoherent because they presuppose intrinsic nature, which is incompatible with dependent origination. He was followed by Aryadeva, Buddhapalita, Bhavaviveka, and Chandrakirti.
Tibetan Buddhism contains several competing interpretations of Madhyamaka. The Gelug school, following Je Tsongkhapa (1357-1419), holds that everything including Buddhahood is empty of inherent existence. The Shentong position, associated with Dölpopa Sherab Gyaltsen (1292-1361) and upheld by non-Gelug schools, holds that ultimate reality is not empty of its own true nature and that Buddhahood is an ultimately real self with infinite qualities. Gorampa (1429-1489) offers an anti-realist reading in which conventional reality is also negated as conceptual fabrication.
The Kaccānagotta-sutta (SN 12.15) is considered one of the clearest expositions of the philosophical Middle Way. It states that the world mostly relies on the dual notions of existence and non-existence, but that someone who truly sees the origin and cessation of the world with right understanding will be free from both notions. The text identifies these extremes as one extreme and then presents the twelve elements of dependent origination as the middle teaching.