Gemara

"Avraham saw... Nitzavim Alav va'Yar va'Yaratz Likrasam" - at first, they stood over him; they saw his pain, and realized that it was not proper to remain there, so they distanced themselves.

87a (R. Yosi): The dots over the letters 'Aleph, Yud, Vov' of "Elav" spells 'Ayo' (where). The angels knew where Sarah was. They asked only because it is proper conduct to inquire about the host's wife.

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.

Question: The angels do understand Aramaic!

(Beraisa): Yochanan the Kohen Gadol heard a Bas Kol (voice from Heaven) in the innermost part of the Mikdash, saying (in Aramaic) that the lambs (Kohanim) that went to battle defeated the Yevanim;

Shimon ha'Tzadik heard a Bas Kol in the Mikdash. It said (in Aramaic) that the army destined to attack the Mikdash was canceled, and the king of the Yevanim died, and his decrees were canceled. They checked and found that he died just when Shimon ha'Tzadik heard the Bas Kol.

Answer #1: A Bas Kol is unlike angels (it is wont to give messages in various places, which mandates knowledge of all languages).

Answer #2: Gavri'el said those voices, for he knows Aramaic;

Gavri'el taught Yosef the 70 languages before Yosef came in front of Pharaoh.

Shabbos 12b (Rabah bar bar Chanah): Sometimes, when R. Elazar visited a Choleh he said 'Hashem should bring Shalom upon you.' Sometimes he said 'Rachmana (the merciful one, in Aramaic) should guide you to Shalom.'

Question: Rav and R. Yochanan taught that one may not pray for his needs in Aramaic, for the angels do not deal with his prayer!

Answer: This is permitted in front of a Choleh, for the Shechinah is with him:

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): Why do women everywhere pray in other languages? 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 taught that an individual may not 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 it is lowly in their eyes.

Gra (OC 101:12): A proof of this is the Yerushalmi cited in Tosfos Rosh Hashanah 16b (DH v'Iyun. The angels knew when R. Chiya had intent in his prayer.) However, the Zohar (Vayera 101b) says that in this world they know only what is given to them to know. The Bavli says that the angels knew where Sarah was, but it agrees that angels do not know people's thoughts. 'They don't recognize Aramaic' connotes that they do not understand it.

Hagahos (in Shulchan Aruch ha'Shalem, 17): The Chida explains the Zohar to mean that when sent to this world they know only what is given to them to know, but when they are in Shamayim they know people's thoughts.

Ma'adanei Yom Tov (Berachos 2:6): What is the source that angels know thoughts? We find Rivkah and Mordechai knew others' (Esav's and Haman's) thoughts through Ru'ach ha'Kodesh. Prophecy through an angel is a higher level, and surely all the more so an angel himself knows people's thoughts! However, Shlomo said "You (Hash-m) alone know man's heart' (Melachim 1:8:39), and it says (Yirmeyahu 17:9,10) "who knows crookedness of the heart? I, Hash-m, who scrutinizes the heart and kidneys..."

Maskil l'David (on Rashi Bereishis 18:2): Rashi says that the angels knew that Avraham wanted to greet them, i.e. this is why he was sitting outside. They do not know people's thoughts - "Hash-m alone discerns man's heart." How can Tosfos say otherwise? Mishnas Chachamim says that angels know thoughts in the heart, but only Hash-m knows thoughts that begin in the kidneys. However, verses teach that even the heart is known only by Hash-m!

R. Akiva Eiger on Gitin, Likutim Shabbos 12b - they know thoughts in the brain, but not in the heart

Note: In Zecharyah 4:5 (in the Haftorah for Chanukah; also 4:13), the angel did not realize that Zecharyah did not understand the olives on the Menorah.

Rashi (Shabbos 12b): Normally, angels must bring a Tefilah in front of Hash-m.

Ra'avad (Tamim De'im 186): The angels understand other languages, all the more so Aramaic, which is close to Hebrew. They do not deal with it, lest Yisraelim pray in Aramaic and Amei ha'Aretz not learn Hebrew. The Gemara thought that a Bas Kol is an angel talking to itself. It answered that it is not an angel. Alternatively, if it is an angel, it is Gavriel. Alternatively, angels know only Hebrew. If so, they do not bring to Hash-m prayers of women who pray alone in other languages. However, one who prays with tears or due to affliction does not need angels, for these gates are never locked. Any Tefilah with great Kavanah is heeded even without angels. Menasheh prayed in Hebrew, and Hash-m accepted it against the angels' attempts to block his prayer (Devarim Rabah 2:20)!

Poskim

Shulchan Aruch (OC 101:4): Some say that even an individual praying for his own needs may pray in any language except for Aramaic, e.g. a Choleh or some affliction in his house. A Tzibur can pray in any language because it does not need advocates.

Mishnah Berurah (19): In a Tzibur, even Arami is permitted, e.g. Yekum Purkan. One who prays at home may not say it.

Eshel Avraham (7): Parshas Derachim says that an individual in a Beis ha'Keneses may pray in Aramaic, for the Shechinah is there.

Kaf ha'Chayim (20): Bitul Chametz and Kaddish were enacted in Aramaic, lest angels understand it and contest it.

Ma'adanei Yom Tov (Berachos 2:7): Angels despise Aramaic because it is a corruption of Hebrew. The Rambam says the same about Arabic!

Shulchan Aruch (YD 335:5): When one prays for a Choleh in front of him, he may pray in any language. If he is not in front of him, he prays only in Lashon ha'Kodesh.

Taz (4): Why does the Tur require (not in front of him) only Lashon ha'Kodesh, unlike the Rosh, who allows any language other than Arame'ic? Since a Choleh needs more mercy, he is stringent for all opinions.