Gemara

16a (Beraisa - R. Meir): Everyone is judged on Rosh Hashanah; the decree is sealed on Yom Kipur;

R. Yehudah says, everyone is judged on Rosh Hashanah; the decree for each matter is sealed in the appropriate time;

The decrees for grain, fruits, water and a person are sealed on Pesach, Shavu'os, Sukos and Yom Kipur, respectively.

R. Yosi says, man is judged every day;

R. Noson says, he is judged every moment.

Question (Rav Yosef): Nowadays we pray concerning sickness and for Chachamim (who are weak). Like which Tana is this?

Answer #1: It is like R. Yosi (or R. Noson).

Answer #2: This is like all the Tana'im, like R. Yitzchak taught;

(R. Yitzchak): Prayer helps both before and after a decree.

16b (R. Yitzchak): Four things tear up decrees against a person - prayer, changing one's name, changing one's deeds, and Tzedakah;

Some say, also changing one's place helps.

17b (R. Yochanan): Teshuvah is great, for it tears (harsh) decrees against a person - "...Va'Shav v'Rofa Lo".

Question (Rav Papa): Perhaps this is before the decree!

Answer (Abaye): "V'Rofa Lo" connotes that he needs to be cured, i.e. there is already a decree.

Question (Beraisa): If one repents during this time (the 10 days of Teshuvah), he is forgiven. If not, even if he brings all the choice rams in the world, he will not be forgiven.

Answer: Teshuvah of the Tzibur tears decrees at any time. Teshuvah of an individual tears decrees during the 10 days of Teshuvah.

Question (Beraisa): "Einei Hash-m Elokecha Bah" - sometimes for good, and sometimes for bad;

A case of for good is if Yisrael were utter Resha'im on Rosh Hashanah and a year of little rain was decreed and they did Teshuvah. They cannot receive much rain, for there is already a decree. Rather, Hash-m brings rain in the right time, where it is needed;

A case of for bad is if Yisrael were utter Tzadikim on Rosh Hashanah and a much rain was decreed and they ceased to be Tzadikim. They must receive much rain, for it was already decreed. Rather, Hash-m brings rain in the wrong time, where it is not needed.

If decrees against the Tzibur can be torn, when they repented the decree should be torn!

Answer: Here it is not necessary, because the same decree can be fulfilled in a good way.

Question (Beraisa) Contradiction: It says "Asher Lo Yisa Panim...", and it says ''Yisa Hash-m Panav Elecha"!

Answer #1 (R. Yosi ha'Kohen): Hash-m is partial to one who sinned against Him, but not to one who sinned against people (and had not yet appeased them).

Answer #2 (R. Akiva): Hash-m is partial before a decree, but not after a decree.

Answer: This refers to decrees against an individual.

Tana'im argue about whether or not a decree is torn:

(Beraisa): The same capital charge was made against two people. One was saved, and the other was not. Both of them prayed. Why was only one answered?

Answer #1 (R. Meir): Only one of them prayed with intent; the other did not.

Answer #2 (R. Elazar): There was already a decree (from Hash-m) against the one who was not saved.

Question: A decree against the Tzibur is not torn!

Contradiction: It says "Kabsi me'Ra'ah Libech", and it says "Ki Im Techabsi...Nichtam Avonech Lefanai"!

Suggestion: Teshuvah helps before the decree, but not after!

Answer: No, it helps for a decree without an oath, but not for a decree with an oath.

Yevamos 105a - Question: "... That is written in truth" - is there writing (in Shomayim) which is not truthful?!

Answer: "That is written in truth" refers to a decree about which Hash-m swore; it cannot be repealed. A decree He did not swear about can be repealed.

(Rav Shmuel bar Ami): A decree with an oath is not torn up - "Therefore I swore to the house of Eli, their sin will never be atoned with animal or flour offerings".

Question (Rav Shmuel bar Unya): What is the source that a decree against a Tzibur is not sealed?

Objection: This cannot be! It says "Ki Im Techabsi...Nichtam Avonech Lefanai"!

Correction: Rather, what is the source that even if a decree is sealed, it can be torn up?

Answer: "Mi ka'Shem Elokeinu b'Chol Kor'einu Elav".

Contradiction: It also says "Dirshu Hash-m b'Himatz'o" (implying that we do not always find Him)!

Resolution: An individual cannot always find Him, but a Tzibur can.

(Rabah bar Avuha): An individual can find Him during the 10 days between Rosh Hashanah and Yom Kipur.

Nidah 70b - Question: It says "Ki Lo Echpotz b'Mos ha'Mes," and it says "Ki Chafetz Hash-m Lahamisam"!

Answer (R. Yehoshua): Hash-m desires to kill sinners who are not repenting, but not those who are repenting.

Berachos 31b (R. Elazar): If one fasts on Shabbos (he suffers greatly, seeing everyone else eating), decrees against him are torn up, even decrees from his birth!

Shabbos 119b (R. Yehoshua ben Levi): If one answers 'Amen Yehei Shmei Raba' will all his Ko'ach (concentration), harsh decrees against him are torn up

Bava Basra 11a (Beraisa): Once, during a famine, a woman and her seven children were in danger of starving. There was no Tzedakah money, so Binyamin ha'Tzadik supported them from his own money. There had been a decree against him; Hash-m tore it up.

Rishonim

Rif and Rosh (1:5): Teshuvah tears decrees against a person - "...Va'Shav v'Rofa Lo". This refers to decrees against the Tzibur without an oath. Prayer helps both before and after a decree. It says "Mi ka'Shem Elokeinu b'Chol Kor'einu Elav", and "Dirshu Hash-m b'Himatz'o". A Tzibur can always find Him; an individual can find Him during the 10 days.

Rambam (Hilchos Teshuvah 2:6): Even though Teshuvah and prayer are always good, they are especially good during the 10 days between Rosh Hashanah and Yom Kipur - "Dirshu Hash-m b'Himatz'o". This refers to an individual. A Tzibur that does Teshuvah and prays wholeheartedly is always answered - ",,,b'Chol Kor'einu Elav".

Question (Tosfos ha'Rosh Nidah 70b DH Kan): Here the Gemara connotes that Teshuvah helps even for an individual. In Rosh Hashanah (17b) we learned that only a Tzibur's decree can be torn!

Answer (Tosfos ha'Rosh): Our Gemara holds like R. Yitzchak, who argues and says that an individual's prayer always helps.

Questions: How can R. Yitzchak explain "Dirshu Hash-m b'Himatz'o"? The Rif and Rosh distinguish between decrees against a person and the Tzibur, and between the 10 days of Teshuvah and the rest of the year. How can they also bring R. Yitzchak's teaching?

Answer #1 (to both questions - Ran DH Melamed): Prayer is effective even when a decree cannot be torn; it can lessen the punishment.

Answer #2 (Parashas Derachim p.27): During the 10 days an individual will surely succeed to tear his decree. At other times it is possible, but there is no guarantee.

Aruch l'Ner (Rosh Hashanah 18a DH Hasam): The Rishonim rule like R. Yitzchak, even though R. Yosi ha'Kohen and R. Akiva argue with him, because we pray like R. Yitzchak.

Tosfos (16a DH k'Man): All Tana'im agree that we pray to be healed and for income. R. Yehudah himself explicitly says so! However, only R. Yosi (and R. Noson) says that we pray not to become sick.

Poskim

Rema (YD 335:10): Some say that if someone is sick in the house, one should go to a Chacham to request mercy. It is our custom to bless Cholim in the Beis ha'Keneses, to give a new name, because changing the name can tear the decree.

Shulchan Aruch (OC 288:4): One who had a frightening dream is permitted to fast on Shabbos to tear up the decree.

Mishnah Berurah (56:1): If one answers 'Amen Yehei Shmei Raba' will all his Ko'ach, i.e. concentration, heart and soul, harsh decrees against him are torn up