Avraham Avinu praying on behalf of Sodom beseeched Hashem to spare Sodom should there be fifty tzaddikim among them.  Hashem responds that He will indeed spare Sodom should there be fifty tzaddikim among the population of Sodom. When Avraham saw that there were not fifty tzaddikim living there, he went down to 45 and so for down to 10.  If Avraham decided at the outset that he would continue praying until Hashem would agree to save Sodom on behalf of 10 tzaddikim, why did he not begin with 10?  Would it not have been wiser for Avraham to begin with a lower number and work his way up?  Should he not have asked Hashem first to save Sodom on behalf of 10 tzaddikim, if Hashem were to say no then say, what about 20? 
 
I believe that Avraham Avinu was praying not only for the people of Sodom but he was trying to prevent a potential chillul Hashem from taking place here: "it would be sacrilege to You!  Shall the Judge of all the earth not do justice?" (Bereishit 18:25).  Prayer must come from the heart, Avraham truly felt the great chillul Hashem that would take place should Sodom be destroyed in spite of there being 50 tzaddikim residing there.  At this point, Avraham did not feel that destroying a city with 40 tzaddikim would be such a great chillul Hashem.  After completing the initial prayer, Avraham rises to a higher level and actually feels that a great chillul Hashem would take place if Sodom were to be destroyed with 45 righteous residents.  This process continued down to 10.  Avraham was praying from the depths of his heart. 
 
Chazal point out that Moshe Rabenu describes Hashem as "HaKel HaGadol HaGibor veHaNorah" "the great, mighty, and awesome G-d" (Devarim 10:17).  Along came Yirmiyahu and limited this description to: "the great and mighty G-d" (Yirmiyahu 32:18).  What happened to His being awesome?  Chazal tell us that Yirmiyahu charged: "strangers are croaking in His Sanctuary where are the displays of His awesomeness?" (Yoma 69b).  It would not be appropriate to use the term "awesome" under such circumstances.  Later on we find Daniel and Nehemiah describing Hashem as: "the great and awesome G-d" (Daniel 9:4 and Nehemiah 1:5). What happened to His being mighty?  They claimed "strangers are enslaving His children, where are the displays of His power?" (Yoma 69b) - It would be inappropriate to describe Him as mighty in such a situation.  Along came the Anshei Knesset Hagedola and announced: "on the contrary! this is His magnificent display of strength, for He restrains His will in that He shows a long-suffering countenance to the wicked, and these are indeed the great displays of His awesomeness because if not for the awe of the nations for the Holy One, Blessed is He, how could one solitary nation survive among the seventy nations of the world?" (ibid.). 
 
Similarly it says in the Gemara: "'who is like You among 'eilim' - the mighty ones,
Hashem' (Shmot 15:11) - who is like You among 'ilmim' - the mute ones? (in Hebrew the spelling is almost identical) (Gittin 56b). According to Chazal Hashem's might and awesomeness is specifically manifested in His hearing the other nations cursing Him, yet remaining silent. The Anshei Knesset Hagedola therefore instituted the words "the great, mighty, and awesome G-d" as the opening of the Shmone Esrei prayer.
 
The Gemara questions the actions of Yirmiyahu, Daniel, and Nehemiah: "now the Rabbis, how did they act thus and abolish the institution that Moshe instituted?" (ibid.).  The Gemara's response is: "because they knew about the Holy One, Blessed is He, that He is truthful, they therefore would not speak falsehood to Him" (ibid.). Hashem is a G-d of truth, and therefore desires that our prayers be truth.  Of course, if the Torah refers to Hashem as "the great, mighty, and awesome G-d", then it must be true. 
 
Yirmiyahu, Daniel, and Nehemiah, although they knew these descriptions to be true, because of the circumstances of their times they could not truly express it from the heart and felt that prayers using these descriptions would not be authentic. It is because "they would not speak falsehood against Him", that they were not prepared to recite words by rote simply "because they are written in the siddur" (or in their case written said by Moshe Rabenu in the Torah)?
 
We do not possess this quality of truth that the prophets had.  We recite "the great, mighty, and awesome G-d" whether we feel it or not, for that is the text that the Anshei Knesset Hagedola established for us.  The prophets, however, were not satisfied with reading from the siddur - they followed the dictates of their heart.
 
The answer is that Avraham's prayers were dictated by his heart - as were those of Yirmiyahu, Daniel, and Nehemiah.  At the beginning Avraham was able to feel and identify with the great chillul Hashem that would be caused by the destruction of fifty righteous people.  At a certain point his fervent prayer elevated him to an even higher level, so that he was able to feel the chillul Hashem that would be caused by the deaths of forty five righteous people.  From his tefilla he was able to rise higher and higher until the potential for chillul Hashem that the deaths of ten righteous inhabitants of Sodom would cause pained him such that it spurred him to try to pray in attempt to avoid it.  When he first began beseeching Hashem, his spiritual level was not such that he would feel the chillul Hashem caused by the deaths of ten righteous people, and a prayer on their behalf would not have been from the heart, while the prayers of the prophets need always be very sincere.
 
Following the chet haegel, Moshe Rabenu prayed to Hashem not to punish the Jewish people.  Moshe reasoned: "why should Egypt say the following: 'with evil intent did He take them out, to kill them in the mountains and to annihilate them from the face of the earth?'" (Shmot 32:12).  Moshe Rabenu was worried purely about the potential chillul Hashem. The same thing happened following the chet hameraglim: Moshe Rabenu asked Hashem not to punish the Jewish people: "then Egypt from whose midst You brought up this nation with Your people will hear ... yet You killed this people like a single man! Then the nations that heard of Your fame will say: 'because Hashem lacked the ability to bring this people to the Land that He had sworn to give them, He slaughtered them in the wilderness" (Bamidbar 14:13-16).  We find a similar reaction from Yehoshua as well "the Canaanite and all the inhabitants of the land will hear and will surround us and cut off our name from the earth.  What will You do for Your Great Name?" (Yehoshua 7:9).  As with Moshe Rabenu, Yehoshua was concerned first and foremost about the potential Kiddush Hashem.
 
Real and sincere tefilla comes purely from the heart.  We pray according to the text set for us by Chazal, however we may add our own personal requests as well - if we wish a refuah shlema than we add it refaenu, and so forth.
 
Chazal derive from Iyov that "Anyone who makes the Name of Heaven a partner in his distress - they double his livelihood for him" (Brachot 63a).  How does one "make the Name of Heaven a partner in his distress"? It would appear to me that when a person davens in an attempt to alleviate his suffering, he must realize that Hashem is suffering with him.  When the Jewish people are in pain, so is Hashem: "At the time a person suffers, what expression does the Shchina articulate: I am burdened by My head; I am burdened by My arm" (Sanhedrin 46a).  The Divine Presence suffers when a person is in pain, it suffers not only when the righteous are in pain, but the wicked as well - including those who have violated transgressions punishable by death at the hands of Beit Din (see Gemara there).
 
It is with this in mind that we recite in our daily Shmone Esrei: "for You are G-d, King, the faithful and compassionate healer". What do we mean when we say He is compassionate?  That He is suffering together with the sick.  We therefore ask: "heal us Hashem - then we will be healed; save us - then we will be saved, for You are our praise, bring complete recovery for all our ailments", so that He will no longer
have to endure our suffering.  We are of course forbidden to speak this way of Hashem, but what we have just said is rooted in Chazal.  The Gemara tells us: "there are three types of people whose lives are not lives: those who are overly compassionate, those who are easily angered, and those who are highly delicate" (Pesachim 113b).  One cannot live a normal life if he is too full of mercy for others.  One moment he hears of one tragedy, the next moment another - he has no peace and tranquility.  It is in this way that Hashem feels our troubles, so to speak.  For each and every Jew that suffers, Hashem suffers as well. We must pray for an end to Hashem's suffering (see Nefesh Hachayim Shaar 2, perek 12).
 
We, in general, first and foremost pray for our welfare - that we and our families should enjoy good health.  After that, we may pray on behalf of other Jews.  Our prayers do not stem from a worry about the honor of Heaven.  In spite of all this, we must do our utmost to "make the Name of Heaven a partner in our distress".  As mentioned above, the text of the tefillot, as established by the Anshei Knesset Hagedola, contain this idea of feeling Hashem's pain.  Is it not therefore preferable for us to actually feel His pain and not simply make mention of it?  Praying while feeling for Hashem's suffering, would be true avoda shebalev - a service of the heart, referring to tefilla (see Rambam Hilchot Tefilla 1:1).  One who is not on this level and who only prays for his own welfare is also serving Hashem.  The mere fact that he turns to Hashem for help and not to Netanyahu, the left, or America, is a clear indication that he knows where to turn for help.  One who needs to be healed, may need to go to the doctor, kuppat cholim, etc, but in the final analysis Hashem is the healer.
 
Tefilla is avoda shebalev - a service of the heart.  By declaring that onlyHashem is able to help us, we are serving Him.  We realize that Hashem wishes for all Jews to be healthy, have parnassa, and only good things.  We find an additional point in Akdamus - if we pray for all these things in order to help us learn Torah then it is more likely that our tefillos will be accepted.  With worries about our livelihood, without wisdom we are unable to learn Torah.  Hashem wants us to learn Torah, by praying for assistance in learning Torah we are praying that Hashem helps us fulfill His will.
 
We must realize that kavana in tefilla is not just a side matter, but is an integral part of our tefilla.  Although the halacha does not require us to repeat tefilla for lack of kavana (perhaps with the exception of the first pasuk of the Shma) - prayer comes from the heart not from the mouth.
 
The Chovot HaLevavot uses the following example: there was a wealthy man who invited the king to lodge at his house.  He instructed his wife and children to make sure the king had everything he may need - food, linens, not to spare anything.  The man, however, went away for his business.  The king arrived and was angry - the host did not even bother to greet the king, his own affairs took precedence.  When we pray, our entire body greets Hashem - we bow, we take three steps back.  But the host, the heart, must be there as well.  The King is not as happy if the heart does not bother to come.
 
One of my rebbeim wondered how people can pray "veten tal umatar" and go outside and claim "oh no, its raining".  I don't find this inconsistent, after all the person acknowledges the bracha of rain, but he would rather have it fall on the fields and not on his head.  However, a man sitting in America praying for the ingathering of the exiles but having in mind "not now, let me first finish university", let the others come first, that is a prayer which is not fully from the heart.  When uttering such a prayer, a person must want to go to Eretz Yisrael himself.  Everyone sincerely prays for health and parnassa, but how many say the words praying for wisdom thinking "I have enough wisdom, I don't need more", or the bracha for Hashem accepting our tshuva while thinking "I have no need for this".  What about "slach lanu" while thinking - thank G-d I have no aveirot.  A person should pray for himself as well as for others.  We all need sechel in order to become gedolei Torah.   We also need parnassa, chochma, tshuva, and everything else.  We must have in mind not only for ourselves but because Hashem wants to give to us, we are praying for His sake as well.