Kaddish is not the name generally given to this prayer in the Talmud. It is referred to there as "May His great Name...", because the congregational response beginning with these words is the heart of this prayer. Only in one minor tractate, (Maseches Soferim 16, 19, and 21) is it referred to as Kaddish.
Composed primarily in Aramaic, the language spoken and understood by most Jews in the Second Temple period (See Tosafos to Maseches Berachos 3a), nevertheless, several Hebrew phrases, which were familiar even to the unschooled, were also incorporated in the prayer.
This particular form of Kaddish d'Rabanan, is so named because of the insertion of the paragraph which begins with the words "Upon Israel and upon our Sages, etc." This Kaddish is recited following the study of Talmud or Midrash and invokes God's blessings on the Sages of Jewry, the transmitters of the Oral Torah.
יִתְגַּדַּל וְיִתְקַדַּשׁ שְׁמֵהּ רַבָּא Exalted and sanctified1These opening words refer to the fulfillment of God’s prophecy, through Ezekiel (38:23) “And I will thus exalt Myself and sanctify Myself; and I will be known in the eyes of the many nations, and they will know that I am Adonoy.”—Levush, Siddur HaGra. One should be very careful to emphasize the letter gimel (ג) when saying the word “Yisgadal (יִתְגַּדַּל),” in order that it not be heard as “Yiskadal (יִתְקַּדַל),” from the word קְדָל which means neck.—Mishnah Berurah 56:2 be His great Name
בְּעָלְ֒מָא דִּי בְרָא in the world which He created
כִרְעוּתֵהּ according to His will
וְיַמְלִיךְ מַלְכוּתֵהּ and may He rule His kingdom2May His sovereignty be revealed.
וְיַצְמַח פּוּרְקָנֵהּ bring forth His redemption
וִיקָרֵב (קֵץ) מְשִׁיחֵהּ. and hasten the coming of His Mashiach.
בְּחַיֵּיכוֹן וּבְיוֹמֵיכוֹן In your lifetime and in your days,
וּבְחַיֵּי and in the lifetime
דְכָל בֵּית יִשְׂרָאֵל, of the entire House of Israel,
בַּעֲגָלָא וּבִזְמַן קָרִיב speedily and in the near future3We pray that the redemption will come speedily and that the process of the redemption itself will not be prolonged.—Siddur HaGra—
וְאִמְרוּ אָמֵן and say Amein.
יְהֵא שְׁמֵהּ רַבָּא מְבָרַךְ May His great Name be blessed
לְעָלַם וּלְעָלְ֒מֵי עָלְ֒מַיָּא forever and for all eternity.4This phrase, the heart of the Kaddish, is found almost verbatim in Daniel 2:20.
יִתְבָּרַךְ וְיִשְׁתַּבַּח Blessed and praised,
וְיִתְפָּאַר וְיִתְרוֹמַם וְיִתְנַשֵּׂא glorified, and exalted and uplifted,
וְיִתְהַדָּר וְיִתְעַלֶּה וְיִתְהַלָּל honored and elevated and extolled5The ten expressions of praise in the Kaddish refer to the ten utterances by which God created the world. (See Maseches Avos 5:1.)
שְׁמֵהּ דְקוּדְשָׁא, בְּרִיךְ הוּא be the Name of the Holy One, blessed is He;
לְעֵֽלָּא (בעשי"ת וּלְעֵֽלָּא מִכָּל) above (Ten Days of Penitence: far above)
מִן כָּל בִּרְכָתָא וְשִׁירָתָא, all the blessings and hymns,6May God be praised and exalted far beyond all the blessings, hymns and praises that are uttered.—Mishnah Berurah in the name of the Vilna Gaon 56:14.
תֻּשְׁבְּחָתָא וְנֶחֱמָתָא, praises and consolations7Consolations are uttered because God is in mourning over the destruction of the Holy Temple and the exile of His people.
דַּאֲמִירָן בְּעָלְ֒מָא, וְאִמְרוּ אָמֵן: which we utter in the world—and say Amein.
עַל יִשְׂרָאֵל וְעַל רַבָּנָן Upon Israel, and upon our Sages,
וְעַל תַלְמִידֵיהוֹן and upon their disciples,
וְעַל כָּל תַּלְמִידֵי תַלְמִידֵיהוֹן, and upon all the disciples of their disciples,
וְעַל כָּל מָאן דְּעָסְ֒קִין בְּאוֹרַיְתָא, and upon all those who engage in Torah study
דִּי בְאַתְרָא הָדֵין in this land
וְדִי בְכָל אֲתַר וַאֲתַר, and in every land.
יְהֵא לְהוֹן וּלְכוֹן May there be unto them and unto you
שְׁלָמָא רַבָּא חִנָּא וְחִסְדָּא abundant peace, favor, kindness,
וְרַחֲמִין וְחַיִּין אֲרִיכִין compassion, long life,
וּמְזוֹנָא רְוִיחֵי וּפוּרְקָנָא ample sustenance and redemption
מִן קֳדָם אֲבוּהוֹן from their Father
דְבִשְׁמַיָּא וְאַרְעָא Who is in heaven and on earth,
וְאִמְרוּ אָמֵן: —and say Amein.
יְהֵא שְׁלָמָא רַבָּא מִן שְׁמַיָּא May there be abundant peace from heaven
וְחַיִּים טוֹבִים and a good life
עָלֵֽינוּ וְעַל כָּל יִשְׂרָאֵל for us and for all Israel,
וְאִמְרוּ אָמֵן: —and say Amein.
עוֹשֶׂה שָׁלוֹם He Who makes peace
(בעשי"ת הַשָּׁלוֹם) (Ten Days of Penitence: the peace)
בִּמְרוֹמָיו in His high heavens
הוּא בְּרַחֲמָיו יַעֲשֶׂה שָׁלוֹם may He, in His mercy, make peace
עָלֵֽינוּ וְעַל כָּל יִשְׂרָאֵל for us and for all Israel,
וְאִמְרוּ אָמֵן: —and say Amein.