The Sanskrit word Nath is the proper name of a siddha sampradaya (initiatory tradition). The word itself means Lord, Protector, or Refuge. The related term Adinath means first or original Lord: a name of Shiva, and beyond that name, the Supreme Reality as originator of all things.
The Nath tradition is a heterodox siddha stream with many sub-sects. It develops from the earlier Siddha or Avadhut sampradaya. Its founding is traditionally ascribed to Shri Dattatreya. As a distinct historical sect it takes form around the tenth century with the fisherman Shri Matsyendranath (also Minanath).
His chief disciples were Caurangi and Gorakshanath. Gorakshanath eclipsed his master in many later branches and is still called the greatest of the ancient Naths. The tradition does not recognize caste. Outcasts and kings both entered it. All branches honor those two. The life, the fish, and the fork toward sannyas are on the Matsyendranath page.
Writers sometimes sort the sampradaya into twelve panths. There were always more sects than that list will hold. The International Nath Order is not one of those medieval Indian monastery panths.
Within the International Nath Order, Nath also names the initiate. Shri Gurudev Mahendranath received the Adinath diksha in 1953 from Shri Sadguru Lokanath, the Avadhut of the Himalayas. In 1978 he founded this Order so the Nath way of life could be lived by householders in the West. Initiates are known as Naths or Devis. They are not bound by the rules of Asian sadhus. Sri Kapilnath stands as Imperator Magus Mundi. The rite by which one becomes a Nath of this Order is initiation. The aims of the Order—Awakening, Siddhi, and a more enlightened social order—are on About. Do not read Nath as another word for Avadhut or for Siddha. Those are attainments. Nath, here, is the name of the stream and of the initiate.