Summary:

The Chabad custom is to recite the Psalm of Ledavid Eilecha during Tachanun with the head over the arm. It must be recited with extreme concentration.

The mystical meaning of the Psalm:[11]

The Psalm of Ledavid Eilecha represents the service of Mesirus Nefesh, to give up one’s life for G-d. It represents the descent of the soul into the depths of impurity, into the clamps of death, in order to remove from there the Divine sparks. One who merits performing Nefilas Apayim properly receives great reward and his enemies fall before him.

Story of Rebbe Eliezer and the power of Tachanun:[12]

Rebbe Eliezer Ben Hurkanis and the Sages once had a dispute whether a certain oven can contract impurity, Rebbe Eliezer held it does not contract impurity while the Sages held that it does. The Sages attacked the position of Rebbe Eliezer with various proofs and Rebbe Eliezer answered each one of these attacks. Rebbe Eliezer concluded that if my position is correct the carob tree will prove it, and the carob tree proceeded to move from its place 100 Ama. The Sages replied that one cannot bring a proof from a carob tree. Rebbe Eliezer continued and said, if my position is correct let the channel of water prove it to you, and the channel of water began to flow the opposite direction. The Sages replied that one cannot bring a proof from a channel of water. Rebbe Eliezer continued and said, if my position is correct let the walls of the Beis Midrash prove it to you, and the walls of the Beis Midrash began to collapse. Rebbe Yehoshua admonished the walls and stated that this is a personal debate amongst Torah Sages regarding a Torah law and you [i.e. the walls] have no business getting involved. The walls stopped falling in honor of Rebbe Yehoshua but remained in a slanted position in honor of Rebbe Eliezer. Rebbe Eliezer continued and said, if my position is correct let the Heavens attest to it and a Heavenly voice proclaimed “What business do you have arguing with Rebbe Eliezer of whom the law is ruled according to his opinion in all places”. Rebbe Yehoshua stood on his feet [and shouted back to the heavenly voice] “The Torah is not in the hands of Heaven”. Eliyahu Hanavi stated that at that point Hashem smiled and said “My children have overpowered me.” That day, they took all the Taharos that were baked in the ovens which Rebbe Eliezer purified and burnt them. In addition, the Sages placed Rebbe Eliezer in excommunication. [The Gemara goes on to tell over how Rebbe Eliezer was informed of his excommunication, and the tremendous damage that was caused to the world as a result of Rebbe Eliezer’s pain.] Rabban Gamliel was the head of the Sanhedrin at that time [and oversaw the excommunication]. Rabban Gamliel’s sister was the wife of Rebbe Eliezer. From the time that this story occurred she would prevent her husband, Rebbe Eliezer, from praying Tachnaun, [as doing so would certainly bring Divine retribution against her brother, Rabban Gamliel]. One day, because she thought it was Rosh Chodesh, she did not prevent her husband from saying Tachanun and he said it. As soon as she found out, she told her husband to get up as her brother, Rabban Gamliel, was certainly killed by his prayer. A proclamation was soon heard in the city that Rabban Gamliel passed away.