While similar in style to the prayers on Shabbos and Yom Tov throughout the year, Shacharis on Rosh Hashana has some significant differences. First, we insert a lot of piyutim, which are special liturgical poems that are not part of the original davening. Second, the tune used by the Chazzan is unique to the high holidays.
The Chazzan begins Shacharis by chanting “Hamelech” in a haunting tune. After singing the first stanza, he makes his way to the Amud and carries on. Upon concluding the Yishtabach prayer, but before beginning the Kadish prior to Barichu, the Ark is opened and Psalm 130 is read. Typically, this psalm is recited responsively, each verse read aloud first by the Chazzan and then repeated by the congregation.
Upon concluding Chazaras Hashatz, the Ark is opened and Avinu Malekeinu – which is reserved for fast days and the Ten Days of Repentance – is recited. It includes multiple pleas to G-d for a good year, forgiveness, good health, redemption, sustenance, and a lot more. A few of the stanzas are slightly altered during the Ten Days of Repentance. For example, we say “Our Father, Our King, inscribe us in the book of life”, while on fast days during the year we read “Our Father, Our King, remember us for good life”. The difference is because we are being judged on Rosh Hashana and the Ten Days of Repentance. As such, we pray for an inscription in the book of the righteous.