Tefillah More than Korbanos
The Gemara relates that Rav Chiya once noticed that Rav Kahana's tefillah was taking especially long. When he finally finished, Rav Chiya questioned him about his lengthy supplications. Rav Kahana explained that due to the sins that the sons of Eli Hakohen they were punished that their descendants would die young. Hashem said: וְלָכֵן נִשְׁבַּעְתִּי לְבֵית עֵלִי, אִם יִתְכַּפַּר עֲוֹן בֵּית עֵלִי בְּזֶבַח וּבְמִנְחָה עַד עוֹלָם - hence I have sworn for the descendants of Eli that even an animal or Minchah offering will for ever not bring them atonement.
Rav Kahana continued to explain that although a korban or Minchah offering will not bring them atonement, tefillah may. Therefore, as a descendant of Eli, he focused intensely during his tefillos and consequently merited a long life. This shows that what a korban could not accomplish, a heartfelt tefillah could achieve.
Through the following story of Chizkiyahu Hamelech we can once again see the immense power of tefillah to nullify even a gezeira that has already been sealed. The Gemara (ברכות ע"א-ע"ב) relays that Yeshayahu Hanavi told Chizkiyahu that it has been decreed that he will die and not have a portion in the world to come because he did not get married and have children. Chizkiyahu asked the navi if there was anything he could do to nullify the decree, and the navi replied in the negative; the decree has been sealed and even if he did get married it was too late to reverse the decree.
Chizkiyahu nonetheless chose not to despair and replied that he learnt from his father that אֲפִלּוּ חֶרֶב חַדָּה מְנַחַת עַל צַוָּארוֹ שֶׁל אָדָם אַל יִמְנַע עַצְמוֹ מִן הָרַחֲמִים - even if a sharp sword is laid on a person's neck, he should not refrain from beseeching (Heaven) mercy. He sent the navi out of the room, turned his head to the wall and started praying and crying intensely. Hashem then answered him through the navi that שָׁמַעְתִּי אֶת תְּפִלָּתֶךָ רָאִיתִי אֶת דִּמְעָתֶךָ הִנְנִי יוֹסִף עַל יָמֶיךָ חֲמֵשׁ עֶשְׂרֵה שָׁנָה. (ישעי' לח ה) - I heard your tefillah and saw your tears hence I am adding to your life (another) fifteen years. Here we see again that even when a gezeira has already been sealed, tefillah has the power to nullify it and even reverse it to a positive decree.
The word שִׂיחַ is often used in reference to tefillah. The gematria of שִׂיחַ is three hundred and eighteen just like the word יֵאוּשׁ with the כּוֹלֵל. (There is a rule in the sefarim regarding the calculation of gematrious; when drawing a parallel between two words which is hinted in their identical numerical value, one word may have one digit more than. This method is known as "the kollel".) One can deduce from this, that tefillah can change even a situation that seems completely hopeless and warrants יֵאוּשׁ despair, in the blink of an eye.
The Nefesh Hachaim says that it is impossible for us to begin to grasp the immense power of tefillah while we are still on this transient world. He quotes the pasuk (עמוס ד יג) מַגִּיד לְאָדָם מַה שֵׁחוֹ - He (Hashem) declares to man what his speech is. This means, explains the Nefesh Hachayim, that when a person is going to stand before Hashem after he completes his years on this world, Hashem will reveal to him the secret of מַה שֶׁחוֹ of how much influence his tefillah had in the upper worlds!