Gemara

32b (Mishnah): Shema Yisrael (may be recited in any language).

"Shema (hear)" - it may be in any language you hear (i.e. understand).

Sotah 33a (Mishnah): Tefilah may be said in any language.

Since it is a request for mercy, one may pray like he desires.

Question: Rav taught that one may not request his needs in Aramaic, and R. Yochanan taught that if one requests his needs in Aramaic, the angels do not deal with his prayer, for they don't recognize Aramaic!

Answer: A Tzibur may pray in any language (Hash-m always accepts its prayer). An individual may not.

Rishonim

Rif and Rosh (Berachos 7a): The Mishnah permits prayer in any language. This is with a Tzibur. Rav forbids praying for his needs in Aramaic. R. Yochanan says that the angels will not deal with the prayer, for they don't recognize Aramaic.

Question (R. Yonah DH Gemara, cited in Rosh Berachos 2:2): We concluded that individuals may pray only in Hebrew. Why do women throughout the world pray in other languages? Since they are obligated to pray, they should pray in Hebrew!

Answer #1 (R. Yonah, citing Chachamim of France): If an individual prays the same prayer as the Tzibur, this is like Tefilah b'Tzibur, and it may be said in other languages. Rav forbids an individual to pray for his needs, e.g. a Choleh (sick person) or some affliction in his house, in Aramaic. One who prays a Tefilah known to the Tzibur can be Yotzei in other languages if he does not know Hebrew. A Tzibur can pray in any language because it does not need advocates. An individual does - "Im Yesh Alav Mal'ach Melitz..." (Iyov 33:23).

Answer #2 (Tosfos Shabbos 12b DH she'Ein, cited in Rosh): Angels know even thoughts of people! Only Aramaic they do not deal with, for they despise it.

Rambam (Hilchos Kri'as Shema 2:10): One can read Shema in any language that he understands. He must be careful about inaccuracies in that language and be meticulous, like one who reads in Lashon ha'Kodesh.

Rebuttal (Ra'avad): This is unreasonable. All other languages are a Perush. Who can be meticulous about the Perush?!

Kesef Mishneh: After "Shema", which permits other languages, it says "v'Limadtem", which mandates Kri'ah Tamah (reading flawlessly). The Rambam requires the best translation of each word into the new language.

Poskim

Shulchan Aruch (OC 62:2): One can recite Shema in any language. He must be careful about inaccuracies in that language and be meticulous, like in Hebrew.

Beis Yosef (DH v'Nikres): The Kolbo says that the Rif allows other languages only in a Tzibur. Angels do not deal with a prayer for one's needs in Aramaic, for they don't recognize Aramaic. I do not understand this. Kri'as Shema does not contain requests for one's needs! There is no reason to distinguish between a Tzibur and an individual. All the Poskim do not distinguish.

Magen Avraham (1, citing Tosfos Sotah 32a DH Kri'as): The Mishnah permits Shema, Tefilah and Birkas ha'Mazon in any language; he is Yotzei only if he understands the language. It omits Hallel and Berachos over Mitzvos and foods, for one is Yotzei them even in a language he does not understand.

Mishnah Berurah (3): He is Yotzei all of these in another language only if he understands it. The ideal Mitzvah is in Hebrew. Nowadays, letter of the law one cannot recite Shema in another language, for we do not know a proper translation for many words, e.g. "Es" and "Totafos". "V'Shinantam" connotes teaching, and also that Divrei Torah should be sharp in your mouth. One who reads Shema in Lashon ha'Kodesh is Yotzei, even if he does not understand. The same applies to Tefilah, Berachos, Hallel...

Bi'ur Halachah (DH Yachol): The Acharonim proved that one is not Yotzei in another language without understanding it. The Pri Megadim was unsure whether one is Yotzei in Hebrew without understanding. In Siman 193, he himself cited the Levush who says that one is Yotzei Berachos without understanding! One must understand the first verse even in Hebrew. (Virtually every Yisrael does.) Even if 'in every language that you understand' does not include Hebrew, one who read without intent was not Yotzei.

Note: Perhaps he must intend for the Mitzvah, but need not understand the words!

Bi'ur Halachah (ibid.): One is Yotzei in another language only if people in the region understand it. Hebrew is an intrinsic language. Other languages are only because countries agreed about them. It is considered a language only if people speak it! 'Charufasi' can make Kidushin only in Yehudah, even if both parties intended for Kidushin (Kidushin 6a). Elsewhere, it is not an accepted language. Tosfos was lenient about Kidush and Birchos ha'Mitzvos only when the person and the people of the region understand the language.

Kaf ha'Chayim (4): There are Kavanos corresponding to the Parshiyos, number of words, initial and final letters that are lost in a translation.

Shulchan Aruch (OC 101:4): One may pray in any language. This is with a Tzibur. Alone, he may pray only in Hebrew. Some say that this refers to a prays for his own needs, e.g. a Choleh. Even an individual may say a Tefilah fixed for the Tzibur in any language. Some permit even an individual to pray for his own needs in any language except for Aramaic.

Magen Avraham (5): If one does not understand Hebrew, it is better to pray in a language that he understands.

Rebuttal (Kaf ha'Chayim 16,17): Anshei Keneses ha'Gedolah fixed a text of Tefilah that makes great Tikunim in the world. One may not change even a vowel. Many Acharonim say that anyone who can pray in Hebrew should do so, even if he doesn't understand it. He should learn the meaning of the words, if possible. One is Yotzei in Hebrew, even if he does not understand it. This applies to Berachos, Korbanos, Pesukei d'Zimra, Birchos Kri'as Shema, Shemoneh Esre, through Aleinu. Other additions (e.g. Piyutim) were not based on secrets of Kabalah, and the Ari Zal did not say them, so one should say them in a language that he understands.

Magen Avraham (6): Hash-m Himself receives Tefilos in Hebrew.

Mishnah Berurah (13): The Chasam Sofer (OC 84,86) proved that prayer in other languages was permitted on occasion, but not on a consistent basis or for the Shali'ach Tzibur. Hebrew would be forgotten! All Ge'onim of the time agreed that it is an absolute Isur. New groups translated the entire Siddur. This led to more Aveiros: they omitted the Berachos of gathering the exiles, and the rebuilding of Yerushalayim.

Mishnah Berurah (16): In front of a Choleh, one may pray in any language, for the Shechinah is there.

A Tzibur can pray in any language because it does not need advocates. An individual does - "Im Yesh Alav Mal'ach Melitz..." (Iyov 33:23).

Kaf ha'Chayim (18): The Heter is only when the Tzibur prays in the other language, but not if it prays in Hebrew.