In the eighth century, Śaṅkara the ācārya of Advaita gathered the wandering renunciates of Bhārata into one ordered brotherhood — ten names, four seats, one Self.
ब्रह्म सत्यं जगन्मिथ्या जीवो ब्रह्मैव नापरः ॥ "Brahman is real, the world is appearance; the jīva is nothing other than Brahman." — the seed-teaching of Advaita Vedānta Attributed to Ādi Śaṅkarācārya
Sannyāsa — formal renunciation — is as old as the Upaniṣads. But by the time of Ādi Śaṅkarācārya (traditionally 8th century CE), the renunciate world had scattered into countless independent streams. Śaṅkara's genius was organisational as much as philosophical: alongside his commentaries on the Upaniṣads, Bhagavad Gītā and Brahma Sūtras, he consolidated the sannyāsīs into ten named orders — the Daśanāmīs — each affiliated with one of four monastic seats (maṭhas) established in the four directions of Bhārata.
Swamiji's own order, Giri ("mountain"), belongs to the northern seat of Jyotirmaṭha in the Himālaya — the direction of tapas, solitude and the unmoving peak.
The tradition recognises an ascending ladder of renunciation — Kuṭīcaka (hut-dweller), Bahūdaka (pilgrim of many waters), Haṁsa (the swan, the wandering meditator) and Paramahaṁsa (the supreme swan, absorbed in Brahman) — with the later literature adding Turīyātīta and Avadhūta for those beyond all outward marks. A Daśanāmī sannyāsī receives a new name at dīkṣā, severs the ritual ties of householder life, and thereafter lives for realisation and the welfare of all beings.
History demanded that some renunciates also bear arms in defence of dharma. From the Daśanāmī body arose the fighting ascetic orders — the Nāgas — organised into akhāḍās ("regiments" or "wrestling grounds"). Of these, the Śrī Pañcadaśanāma Jūnā Akhāḍā is the oldest and largest. The two wings are one body: śastra (the sword) protected śāstra (the scripture), so that the wisdom could reach us intact.