The Rosh Hashanah Liturgy: Structure & Differences from Ordinary Sabbaths

Here's some basic info to help you find your way through the High Holiday prayers, assuming that you aren't totally lost in the ordinary Shabbat service. If you aren't, I'll try to get a post about that up, too. In the meantime, look here.
This is the handout from the class that started the entire series of English classes in prayer at Hod Vehadar (which are gradually being adapted and repeated in Hebrew). 
Page numbers refer to the Rinat Yisrael Ashkanaz (R) and Birnbaum machzorim (B), which are the ones most available at Hod veHadar, but the structure is the same in all traditional machzorim (machzor=high holiday prayerbook). 

Additions to morning and evening services, from Rosh Hodesh Elul through Hoshana Rabba: Psalm 27 (R 43, B 46). “Hope in the Lord; be strong and of good courage! O hope in the Lord!”

Changes in all services from Erev Rosh Hashanah through Neilah (the end of Yom Kippur):

A.       Additions to the first two and last two blessings of the amida, for example: R 34, 39; B 31/2, 37/8.

1.         “Remember us to life, O King who delights in life; inscribe us in the book of life for your sake, O living God.” 

2.         “Who is like You, merciful Father: In mercy you remember your creatures to life.”

3.         “Inscribe all the people of your covenant for a good life.”

4.         May we and all Israel your people be remembered and inscribed before you in the book of life and blessing, peace and prosperity, for a good life and for peace.

B.        Changes in the Kaddish, including the Mourner’s Kaddish (R 43; slightly different in B 45/6):

1.          לעלא ולעלא מכל ברכתא  instead of לעלא מן ברכתא 

2.          עושה השלום   instead of עושה שלום.

C.        There are additional changes to blessings that are said only on weekdays between RH and YK.

In all holiday amida prayers on Rosh Hashanah and Yom Kippur:

The third blessing ( (קדושת השםbegins with three paragraphs beginning with ובכן. (R 35-36; B 31/2-33/4). In the context of declaring God’s holiness we also “reaffirm loyalty to a universal outlook and world brotherhood, the wellbeing of Israel, and the triumph of moral law” (Morris Silverman). The origin of these prayers is unknown but the concluding sectionקדוש אתה ונורא שמך   is found in the ancient Land of Israel rite.  Note the final phrase המלך הקדוש   instead of  האל הקדוש. Emphasis on God’s sovereignty is a major theme of all Rosh Hashanah prayers.

The fourth blessing of Shabbat and holiday amidot is  קדושת היוםthe “sanctity of the day,” and is adapted for the particular day. The Rosh Hashanah version is special because the concluding blessing replaces מלך העולם with מלך על כל הארץ. The difference in meaning is not great but the change adds emphasis. Also in Kiddush.  

Piyutim are elaborate, poetic additions to the obligatory service, dating from the sixth century  through the middle ages.  They are not obligatory; indeed many rabbis opposed them. Ibn Ezra: “When we pray it is forbidden to inject in our prayers piyutim, the basic meaning of which we do not understand.” He lost the battle. There are many variations and customs about which are read on which day and which are skipped. There is no one right way, but many local variants and customs.

Rosh Hashanah Shaharit

A.       The morning blessings and psalms (R 108-153; B 51-170) are unchanged but the hazzan starts formal chanting at “Hamelech” rather than “Shokhen ad” to emphasize the sovereignty theme.

B.        Just before Barechu, Psalm 130 (R 156; B 171-172) is often added. Its theme is forgiveness: “O Israel, put hope in the Lord, for with the Lord there is kindness, with him there is great saving power. He will redeem Israel from iniquity.”

C.        In Rinat Yisrael, there is only one brief addition to the Shema and its blessings section of the service (R 154 [not the first printing]; B 171-72). In Birnbaum, there are many additions that Rinat Yisrael relegates to the appendix (references in teeny letters) and there they stay.

D.       In the silent amida in Shaharit (R 165-170; B 202-210), the changes are as described above.

E.        The reader’s repetition is different on the first (R 171-186; B 210-228, 262-270) and second  (R 279-296; B 230-270) days. In RY on Day 2, after skipping forward, remain. Everything is repeated.

F.         Avinu Malkaynu (R 186-88, 297-298; B 271-76) is recited on Rosh Hashanah morning, Yom Kippur, the intervening weekday mornings and other fast days (but not Tisha B’av). Its origin in the public prayer services in times of drought prescribed by the Mishna and Talmud. Rabbi Akiva is recorded as praying just Our Father, our King we have no King except You… have mercy on us for your sake.” It has since grown and there are now several versions.  Most often, the כתבנו lines in the middle are read repeating after the Hazzan and the last one is sung.

G.       Torah service is substantially the same as Shabbat and other holidays, with changes that emphasize God’s sovereignty. There are five aliyot + maftir (in a second scroll) on weekdays and seven on Shabbat. For commentary on the readings, click here.

H.       After the haftara reading and prayer for the State of Israel but before the Torah scrolls are returned to the ark, it’s Shofar time (round 1), but not on Shabbat. (R 203-205, 310-311; B 315-320). Customarily, 100 shofar blasts are sounded. These are the first 30.

First Psalm 47, is read seven times (“God is gone up amidst shouting, the Lord amidst the sound of the horn. Sing praises to God…  For God is the King of all the earth”) and seven more verses are read responsively. The person blowing the shofar, adds another verse and then recites the mitzvah blessing and sheheyanu. The hazzan reads out the order of the shofar calls. After the last (tekiah gedola) everyone says aloud “Happy are the people who know the sound of the shofar, Lord they walk in your light” (Ps. 89) and modulate into the familiar Ashrei psalm (gliding over two more verses also from Ps. 89 that aren’t in all machzorim).

Rosh Hashanah Musaf

A.       Begins with Hineni, the hazzan’s prayer (R 207; B 326). Sometimes read before the repetition

B.        Nine blessings instead of seven. The three central blessings are:

1.         Malchuyot (sovereignty): God as sovereign (begins with the usual kedushat hayom). (R 211-215; B 331-337)

2.         Zichronot (remembrance): God’s mindfulness of us and our situations. (R 215-218; B 337-342)

3.         Shofarot (the Shofar): God as redeemer. (R 218-220; B 342-346)

4.         Each of these begins with an introductory statement (in Malchuyot after the usual kedushat hayom.  In Malchuyot, this is Aleinu, which was later copied to the end of each service.

5.         The intro is followed by ten verses on the appropriate theme, three from the Torah, three from the Prophets and three from the Writings, and another from the Torah.

6.         Each section ends with a concluding blessing.

C.        Again, the reader’s repetition is different on the first (R 223-250; B 349-358, 361-408) and second  (R 328-351; B 359-408) days.

1.         For many people, the liturgical climax is Un’taneh tokef (R 228-230; 329-331 B 361-363). For others, it’s the most problematic. In either case, it is actually a piyut leading up to the Kedusha. For more info see “Who by Fire, Who by Water” , click on Google Preview, and then buy it if you can. Do "Repentance, Prayer and Tzedakah avert the sever edecree"?

2.         Shofar blowing in the repetition: at the end of Malchuyot, Zichronot, and Shofarot. Each time followed by Hayom Harat Olam and a prayer that our prayers and shofar blowing be accepted.

For more reading: 

Rosh Hashanah Liturgy: God, humanity, and the relationship between the two by Rabbi Dr. Reuven Hammer.

Prayer Marathons Of Rosh Hashanah And Yom Kippur: New strategies for an ancient tradition by Nigel Savage.