Tahanun, meaning "supplication," is a prayer that is recited following the weekday Amidah at Shacharit and Mincha. The source of the prayer is in Daniel (9:3) and I Kings (8:54), where verses indicate that prayer should always be followed by supplication. Based on this, Talmudic sages developed the habit of adding a personal appeal to God following the set prayers. In the fourteenth century, these spontaneous supplications were standardized and turned into the prayer of Tahanun.
On all days except Monday and Thursday, Tahanun is very short. Sephardic and some Ashkenazic congregations preface Tahanun with a short form of vidui, a confessional prayer, and a verse from Exodus (34:6) that describes the thirteen attributes of God. In most Ashkenazic synagogues, Tahanun begins with several introductory verses from II Samuel (24:14) and Deuteronomy (6:4), and then continues with Psalm 6, which King David composed while he was sick and in pain. Sephardic liturgy substitutes Psalm 25 for Psalm 6.
This first part of Tahanun should be said while seated, bent over, with the face lowered on the left forearm. The use of the left arm is reminiscent of the Daily Sacrifice brought in the Temple, which that was laid on its left side to be slaughtered. When a man is wearing tefillin on the left arm, he lowers his face to his right forearm instead. A person's arm should be covered with a sleeve, tallis, or other covering. This posture, developed in the post-Talmudic period, is symbolic of Temple practice, in which people knelt down until their faces touched the ground to show humility and submission to God. The pose was also used by Moses and Joshua, who fell on their faces before God after the sin of the Golden Calf. Because of this practice, Tahanun is also known as nefilat apayim ("falling on the face"). Because Joshua fell on his face before the Ark, one only puts one's head down when praying in front of an Ark containing a Torah scroll. Otherwise, it is proper to sit with the head up.
The second part of Tahanun on these days begins "shomer yisrael" ("O Guardian of Israel"). In it, a person acknowledges that God is his guardian whom he is dependent on, and also stresses reasons why the people of Israel are important to God and deserve God's mercy. This section is recited while sitting in a normal position.
The final paragraph begins "v'anachnu lo nayda" ("We know not what to do"). It emphasizes that the reciter has prayed in every possible way – sitting, standing, and with his heads down. At this point, there is no alternative but begging God to help for the sake of His own glory. This paragraph is composed mainly of verses from the Psalms. The custom is to sit for the first three words of the paragraph, then to stand for the remainder.
On Monday and Thursday mornings, a longer series of supplications, called V'hu Rahum, is added at the beginning of Tahanun. These supplications were written during the Middle Ages and reflect despair and helplessness. The basic theme is that God is the Jewish people's only hope for salvation from the suffering that they have brought upon themselves through sin.
Folowing these entreaties, the first Psalm of the regular Tahanun is recited. Then, a section beginning "Adonai elohei yisrael" (Hashem, God of Israel) is inserted. According to tradition, King Hezekiah composed this section when Jerusalem was besieged by the Assyrians and the situation seemed hopeless (II Kings, ch. 18).
The reason that extra sections are added on Mondays and Thursdays is because, according to tradition, these days are favorable times for God to respond to our pleas. Ever since court sessions began to be held on these days in the Temple era, they have been seen as days of judgment. Also, Moses ascended Mt. Sinai on the fifth day of the week to receive the second set of the Ten Commandments, and he descended on the second day of the week.
Since Tahanun is a prayer of sorrow, it is not said on days with festive characters. It is omitted on: Shabbath, festivals, Rosh Chodesh, Chanukkah, Tu b'Shevat, Purim and Shushan Purim, Purim Katan, the entire month of Nissan, Lag b'Omer, from the first day of Sivan through the day after Shavout (according to some, through the 12th of Sivan), Tisha b'Av, the fifteenth of Av, and from Yom Kippur through the day after Sukkot (according to some, until the end of Tishrei). Tahanun is also omitted from Mincha on the day before any of these days, and from Shacharit on the days before Rosh Hashanah and Yom Kippur. Many congregations also omit Tahanun on Yom Ha'atzmaut, Yom Yerushalayim, and Passover Sheni.
The only one of these days that seems incongruous with the rest is Tisha b'Av, a day of national mourning. It is understandable that the supplications are not recited on festive days, but why not on a day of sorrow? One explanation is that Tahanun is not said in order to show that destruction paves the way for eventual redemption. Others say that Tisha b'Av will become a festival in the Messianic era. Tahanun is also omitted in a synagogue: where a bridegroom is present on the day of his wedding or the seven days following his wedding; when a circumcision is taking place in the synagogue that day; and when the father of the baby, the sandek (who holds the baby during circumcision), or the mohel ( who performs the circumcision) is present on the day of a circumcision. In Hasidic congregations, Tahanun is also omitted on the anniversary of the death of sect's Rebbe (head), since that is considered a day for religious renewal and celebration. Finally, Tahanun is omitted in a service in the house of a mourner, so as not to add to the mourner's grief by highlighting God's judgment.
Sources: Donin, Hayim. To Pray as a Jew: A Guide to the Prayer Book and the Synagogue Service. NY: Basic Books, 1991.
Kolatch, Alfred J. The Jewish Book of Why/the Second Jewish Book of Why. NY: Jonathan David Publishers, 1989
Schermon, Rabbi Nosson, editor. The Complet Artscroll Siddur. New York: Mesorah Publications, Ltd, 1984.