Prayer services on Yom Kippur are lengthy and solemn. The level of public solemnity on Yom Kippur surpasses that of any other festival, including Rosh Hashanah. We pray for forgiveness for sins between ourselves and G-d, and we correct wrongful actions between ourselves and other people.

Most of the holiday is spent in prayer in the synagogue. A machzor prayer book is used to lead us through the special prayer service. We pray Kol Nidrei and Maariv on the evening of Yom Kippur. On Yom Kippur day we recite Shacharit, the Torah Reading, Yizkor, and Musaf. After a short break, we meet again to pray Mincha and Ne'ilah. After Maariv, we can break the fast.

To differentiate the Yom Kippur prayer service from other days and to show the solemnity of the prayers, special melodies are used for some of the prayers and the talit is worn for evening prayers (usually the talit is only worn during the day).  

Kol Nidrei

Kol Nidreiis both the opening prayer and the name for the evening service that begins Yom Kippur. Kol Nidrei means "All Vows." In this prayer we ask G-d to annul all vows we may make to G-d (not to people) in the next year. The prayer is in Aramaic because that was the language of its composers. This prayer should be recited before sunset because vows should not be annulled on Shabbat or holidays. The Kol Nidrei prayer is recited three times. Each time it is recited, the chazan's (cantor's) voice gets louder as if he is respectfully approaching a King with a request. The fact that i opens the Yom Kippur prayer service, and not the fact that it is an annulment of vows, makes the Kol Nidrei service so meaningful.   

Maariv

Maariv on Yom Kippur differs from Maariv on other evenings in that it includes selichot (penitential prayers) and the confessional (the confessional is said ten times on Yom Kippur). When we say "ashamnu" (we have trespassed) and "al chet" (for the sin), it is customary to beat our breasts because it is said our hearts are the source of our sins. On Yom Kippur, "Baruch Shem" (Bless G-d) is said out loud in contrast to the rest of the year when it is whispered. Maariv ends with "Avinu Malkanu" (Our Father, Our King). When evening prayers end, we should maintain a solemn spirit.

Shacharit

Shacharit on Yom Kippur is similar to that of other holidays. The Amidah includes the confessional. The repetition of the Amidah includes piyyutim (poems), selichot (penitential prayers), and the confessional. Shacharit ends with "Avinu Malkanu" (Our Father, Our King). 

Torah Reading

The Torah Portion, Vayikra (Leviticus) 16, describes the sacrifices offered on Yom Kippur in the time of the Temple. In this portion, G-d decrees that the tenth of Tishrei will always be a day of atonement.

Let it be a statute for you forever: in the seventh month, on the tenth of the month, you shall starve your vital energies and do no manner of work.... For on this day it shall bring atonement upon you, to purify you, before G-d shall you become pure of all your aberrations. 
(Vayikra (Leviticus) 16: 29-30) 

Maftir is Bamidbar (Numbers) 29:7-11 which also describes the Yom Kippur day and sacrifice. 

And on the tenth (day) of that seventh month there shall be for you a convocation to the Sanctuary, and you shall let your souls starve; you must do no manner of work. And you shall bring to G-d as an ascent offering to express compliance. (Bamidbar (Numbers) 29:7-8) 

The Haftarah, Isaiah 57:14-58:14, focuses on the purpose of fasting, to induce us to be more righteous in our relations with fellow men. 

And you draw out your soul to the hungry, and an afflicted soul you sate, then your light shall shine in the darkness, and your darkness shall be like noon. (Isaiah 58:10)

Yiskor

Yizkor is the memorial prayer service. It is recited after the Haftarah. It is customary for those who have not lost parents to temporarily leave during the Yizkor prayer service. Yizkor is recited on Yom Kippur because it puts the living in a more solemn mood and because those who have passed away also need to forgiveness. 

Musaf

Musaf is the final part of the Yom Kippur morning prayer service. The Amidah contains the confessional. The repetition of the Amidah includes piyyutim (poems), selichot (penitential prayers), and the confessional. The Yom Kippur Musaf service consists of two parts: the Avodah (description of the Yom Kippur prayers and rituals during the time of the Temple) and the Martyrology (description of the murders of ten talmudic sages tortured by the Romans). Musaf ends with the hymn "Ha'yom Te'amtzaynu" (This day shalt thou strengthen us). 

A short break is usually taken between musaf and mincha. 

Mincha

Mincha begins with the Torah reading, Leviticus 18. This portion focuses on the prohibition of certain marriages and illicit relations. Some say this portion is read because Yom Kippur used to be a day on which young men chose brides, and the reading reminds the young men to choose their brides with purity and family in mind. Others say this portion is read because it follows the Torah portion read in the morning. The Book of Jonah is also read. In Jonah we learn how the people of Nineveh repented and how G-d forgave. After the Torah is returned to the ark, the same Amidah as the one recited during Shacharit is recited. The repetition of the Amidah includes piyyutim (poems) and selichot (penitential prayers). 

Ne'ilah

Ne'ilah (locked) is the concluding service of Yom Kippur. The ark holding the Torah is kept open throughout this service, thus we must stand throughout the service. There is a tone of desperation in the Ne'ilah prayers as this is our last chance to atone before G-d seals our judgment in the books. The service is sometimes referred to as the closing of the gates. The Ne'ilah service ends with a long blast of the Shofar, reminiscent of the Shofar which was blown in the Temple at the end of Yom Kippur.

Maariv

Maariv at the end of Yom Kippur is the same service as at the end of other holidays. After Maariv, it is time to break the fast. The spirit is joyful because we believe that G-d has forgiven us.