There are simple rules regarding the order of our prayers. The Shemona Esrei, or Amida, is the most sacred prayer, and is a form of meditation and connection to G-d.
The weekday Amida, consists of nineteen blessings. They are divided into three categories: three blessings of praise, thirteen blessings of requests, and three blessings of thanksgiving. On Shabbat and holidays, we do not make requests, so that the Amida contains the same three blessings at the beginning and end, but only one blessing describing the sanctity of the day.
The only exception to this rule is on the two days of Rosh Hashanah. Instead of one middle blessing, there are three blessing in the Mussaf service.
These blessings are known as Malchiot, describing G-d as King, Zichronot, where we point out the times when G-d remembered us, and Shofarot, the various times when the Shofar is sounded.
But the main focus of Rosh Hashanah, is the acknowledgement that G-d is King, and all that we have, including every breath we take, depends on Him. This acknowledgement cries out today in the upside down world we live in. Darkness is light, and light is darkness. People are wandering aimlessly, trying to create substitute philosophies that they foolishly believe, will benefit mankind.
It is clear from traditional Jewish sources, that accepting the Kingship of the Al-mighty, is the only thing that will bring light and goodness to the world.
This is why, establishing this Divine Kingship, is the prerequisite for the prayers and celebrations of the coming holidays, to bear fruit. G-d was always the King. He is the King, and will be the King, forever and ever. Shana Tova. Happy New Year.