In Parshas Breishis (2:5), the Torah tells us that no vegetation was growing yet because there was no rain. Rashi quotes the Medrash Breishis Rabbah which explains that Hakodosh Baruch Hu withheld rain until he created Adam Harishon, who would understand the need for rain and offer a tefillah.
There is an old minhag recorded in Shulchan Aruch to recite parshas ha'mon every day. The basis of the minhag is that we should always keep in mind the fact that parnasah comes directly from Hashem. Hashem wants people to daven not because He needs their tefillos but rather because man needs tefillah. When we daven, we recognize that Hashem is in full control of everything and that we have the ability and the right to ask Him to take care of all of our needs.
In the opening chapter in Sefer Shmuel the story is told of Elkonah who had two wives, Peninah, who had seven children, and Chanah, who had no children. The Gemarah (Bava Basra 16A) interprets the passuk to mean that Penina would tease Chanah and aggravate her l'shaim Shomayim to get her to offer tefillos to have children.
In Parshas Vayera (18:17) the Torah tells us that Hashem thought to himself that it is not proper that he should destroy Sodom and Amorah without first notifying Avraham Avinu. In the commentary Pitochei Chosom by Rabbi Abuchazirah, this is understood to imply that Hashem wanted Avraham Avinu to pray for the people of Sodom. Avraham did so, and according to the Midrash, it is only because of the prayers of Avraham that Lot and his two daughters were spared. The daughters of Lot thought that the entire world was destroyed, they and their father were the only survivors, and for that reason they thought that they had to act to preserve mankind. But the Torah tells us explicitly (19:29) that they were only spared in the zechus of Avraham. Avrohom's tefillos did not help to save the rest of the people who lived in the five cities, but his prayers did accomplish saving Lot and his two daughters.
Sometimes Hashem will see to it that an individual should be elected to an official capacity in order that we should realize that we all have a need to daven. Throughout our lives we ought to daven for what we need - single people have to daven that they get the proper shidduch (Tehilim 32:6), married people have to daven for healthy children, etc. And now we should all daven that government officials should be given the wisdom to treat the Jewish people properly.