The Unetanneh Tokef is recited in the Musaf, or Mussaf, service on Rosh Hashanah.
Unetanneh Tokef is a medieval Jewish liturgical poem about the awe of divine judgment. In the Ashkenazi rite, it is inserted during the repetition of the Musaf Amidah, just before the Kedushah. Musaf is the additional prayer service for Shabbat and festivals, historically connected with the additional Temple offerings associated with those days.
The poem is especially famous for its dramatic description of the heavenly court, the opening of the Book of Remembrance, and the inscription and sealing of human fates. These themes fit the central Rosh Hashanah Musaf blessings of Kingship, Remembrance, and Shofarot.
The answer is not Maariv, the evening service, or Kabbalat Shabbat, a Friday-night service. Practice does vary by community: some Sephardic congregations omit the poem, while others recite it before or around Musaf. It is also commonly heard on Yom Kippur in Ashkenazi communities, so it is not exclusive to Rosh Hashanah.