The phrase nowruz is first attested in Middle Persian as nog ruz; it has its root in an ancient Iranian tradition of seasonal festivals (gahanbars). Younger Avestan material dating to, perhaps, at its earliest about the mid-1st milennium BC (Yashts and the Afrinagan-i Gahanbar, a [later] liturgical text), allude to six such festivals: mid-spring, mid-summer, harvest, homecoming of the herds, and "mid-year", referring to winter. The last festival mentioned is Hamaspath-Maedaya, the meaning of which is uncertain. These were arranged in a typically Zoroastrian "6+1 scheme" by associating them with elements of creation, and thus divinities. The aforementioned six were associated respectively with the sky/Power, water/Health, plants/Immortality, animals/Good purpose, and Humans/Wisdom. This leaves out Fire and the final member of the divine heptad Asha, which according to Jenny Rose in the Wiley-Blackwell Companion was incorporated into the seventh and holiest festival, that of Nowruz.
The association with Asha allows us to understand its religious significance - Asha is the natural order of things, harmony, order, and law. Celebration of asha during the new year makes sense, to ensure an orderly passing from one year into the next. The first allusion attested in a surviving tradition seems to be the spring festival at the beginning of the epic Vis and Ramin, which was written down in the 11:th century AD but dates back about a thousand year before that. It appears that a first attempt to fix it was made in the early 6th century - numerous calendar changes since the pre-Achaemenid period has moved the liturgical calendar out of sync with the secular one. This placed it at the spring equinox (the return of light), but not at the calendral new year - this only happens as late as the Seljuq's, with the solar calendar of Jalal al-Dawla.
By notes in the Middle Persian Nerangestan, we can infer that Gahanbars were religiously important in the strict sense, in that failing to observe them would count against you when passing the Bridge of Reckoning into the afterlife. We find a lighter side to it as well however in the writings of al-Biruni, who notes that stories such as that of the mythic ruler Jamshid's discovery of sugar were told during the festival. al-Biruni says that whereas Nowruz was associated with creation, the winter festival of Mehragan was associated with resurrection of that creation.
While Nowruz and other festivals have been celebrated continuously by Zoroastrians, Rose notes that the immense popularity and uniformity of Nowruz is a 20th century phenomenon, the result of the Pahlavi dynasty's effective use of mass media and appeal to Iranian tradition, as well as post-1979 reactions and counter-reactions. Its celebration is certainly well attested throughout the Islamic era, but it appears that it was only "standardized" in the mid-20th century (to some consternation for observant Zoroastrians, who obviously wish to maintain its religious significance).
Further reading: Apart from the aforementioned chapter by Rose, Encyclopaedia Iranica has thorough (if somewhat tedious) articles on Nowruz and its history.