The Shulhan Aruch (Orah Haim 122:2) writes that immediately upon completing the recitation of the Amida, one should recite the verse, "Yiheyu Le’rason Imre Fi Ve’hegyon Libi Lefanecha Hashem Suri Ve’go’ali" (Tehillim 19:15). The literal translation of this verse is, "May the words of my mouth be pleasing and the thoughts of my heart before You, Hashem my Rock and my Redeemer."
This verse should be recited twice – or, in some instances, three times – at the end of the Amida. First, one should recite "Yiheyu Le’rason" immediately after the final Beracha of the Amida – "Ha’mebarech Et Amo Yisrael Ba’shalom Amen." Thereafter, it is customary to recite the prayer of "Elokai Nesor," which, as the Gemara records, was a private prayer recited by the Sage Mor Bereh De’Ravina. With time, it became a standardized prayer text that is recited after the Amida. After one completes "Elokai Nesor," he should repeat "Yiheyu Le’rason." The Mishna Berura (Rabbi Yisrael Meir Kagan, 1839-1933), citing the Hayeh Adam (Rabbi Abraham Danzig of Vilna, 1748-1820), writes that it is proper to recite additional prayers after the Amida, as well. Specifically, one should pray for his livelihood and that his children and descendants should be God-fearing and devoted students of Torah. (A number of Siddurim include texts for such prayers.) It is proper to recite these supplications after "Elokai Nesor" (and the second recitation of "Yiheyu Le’rason"), and one who does so should then recite "Yiheyu Le’rason" a third time after this prayer.
The Mishna Berura adds that the scholars of Kabbala (Jewish mysticism) afforded great importance to the recitation of the verse "Yiheyu Le’rason." This verse contains ten words, which correspond with the ten Sefirot ("emanations" with which, as Kabbalistic thought teaches, God created the world). Furthermore, the letter "Yod" – which has the numerical value of ten – appears ten times in this verse, and in fact this verse both begins and ends with this letter. Additionally, this verse contains forty-two letters, alluding to the forty-two-letter Name of God.
Thus, the Mishna Berura emphasizes, one should make a point of reciting "Yiheyu Le’rason" slowly and with concentration, as proper recitation of this verse has a significant impact upon the acceptance of one’s prayer.
The Hazan should also recite the verse "Yiheyu Le’rason" upon completing the repetition of the Amida, before proceeding to the next part of the service.
Finally, the Shulhan Aruch mentions in this context (122:3) the importance of reciting after the Amida the phrase, "Aseh Lema’an Shemach, Aseh Lema’an Yeminach, Aseh Lema’an Toratach, Aseh Lema’an Kedushatach" ("Act for the sake of Your Name, act for the sake of Your right [hand], act for the sake of Your Torah, act for the sake of Your sanctity"). One who customarily recites this prayer, the Shulhan Aruch writes, will be privileged to behold the Shechina (Divine Presence). This indeed appears in our Siddurim after the Amida, and one should ensure to recite it with concentration.
Summary: One should recite the verse of "Yiheyu Le’rason" after the Amida immediately following the Beracha of "Ha’mebarech Et Amo Yisrael Ba’shalom," and then again after "Elokai Nesor." Those who add other supplications after the Amida should recite "Yiheyu Le’rason" a third time after those supplications. This verse should be recited slowly and with intense concentration.