The special prayer book used on Rosh Hashanah and Yom Kippur is the machzor.
A machzor is a festival prayer book containing the distinctive liturgy of Judaism’s High Holy Days. It includes the regular framework of synagogue prayer along with holiday additions, including piyyutim (liturgical poems) and passages focused on divine kingship, remembrance, judgment, repentance, and the shofar.
The word machzor comes from a Hebrew root meaning “to return” or “cycle.” Historically, it could refer to a prayer book for the entire annual cycle. Over time, Jewish communities increasingly used siddur for weekday and Shabbat prayers, while machzor came to mean a book for festival services.
A common mix-up is confusing the machzor with the siddur. A siddur is the general prayer book used throughout the year; a machzor is a specialized holiday edition. Haggadah belongs specifically to the Passover seder, while Tanakh refers to the Hebrew Bible.