The machzor is the special prayer book used on Rosh Hashanah to lead worshippers through the special Rosh Hashanah prayer service. The main themes of the prayer service are repentance by man and judgment by God, Our King.
Rosh Hashanah Torah Readings: Day One
On the first day, we read Beresheet (Genesis) XXI. This Torah portion tells of the birth of Isaac to Abraham and Sarah. According to the Talmud, Sarah gave birth on Rosh Hashanah. The haftara for the first day of Rosh Hashanah is I Samuel 1:1-2:10. This haftara tells the story of Hannah, her prayer for offspring, the subsequent birth of her son Samuel, and her prayer of thanksgiving. According to tradition, Hannah's son was conceived on Rosh Hashanah.
Rosh Hashanah Torah Readings: Day Two
On the second day, we read Beresheet (Genesis) XXII. This Torah portion tells of the Aqedah where Abraham almost sacrificed his son Isaac. The sounding of the shofar is connected with the ram sacrificed instead of Isaac. The haftara for the second day of Rosh Hashanah is Jeremiah 31:1-19. This portion mentions God's remembrance of His People. On Rosh Hashanah we need to mention God's remembrances, thus this portion fits the day.
Rosh Hashanah Maftir
On both days, the Maftir is Bamidbar (Numbers) 29:1-6.
"And in the seventh month, on the first of the month (aleph Tishrei or Rosh Hashanah), there shall be for you a convocation to the Sanctuary; you must not do any service work."
The portion goes on to describe the offerings which our ancestors were obligated to make as an expression of compliance to God.
Before, during and after prayer services, we say to others "Shana Tova V'Chatima Tova" which means "good year and good seal in the Book of Life."