Why pray? There are four schools of thought.
The simple idea of prayer: the social approach
One confronts God and asks for one’s needs, and God really does listen to prayer. This is the primary approach to prayer found in the Jewish Bible and by Chazal, the sages of the Mishnah and Talmud.
“The concept of prayer is based on the conviction that God exists, hears, and answers – that God is a personal deity. In a sense it is a corollary of the biblical concept that man was created ‘in the image of God’, which implies, inter alia, fellowship with God. Although prayer has an intellectual base, it is essentially emotional in character. It is an expression of man’s quest for the Divine and his longing to unburden his soul before God….That prayer is answered is an accepted biblical verity; but Scripture is no less emphatic that not all prayers are answered.” – Encyclopaedia Judaica, Prayer
The rationalist approach
In this view, ultimate goal of prayer is to help train a person to focus on divinity through philosophy and intellectual contemplation. This approach was taken by Maimonides Samuel Ibn Tibbon, Joseph Albo, and the other medieval rationalists.
The educational approach
In this view, prayer is not a conversation. Rather, it is meant to inculcate certain attitudes in the one who prays, but not to influence. This has been the approach of Rabbenu Bachya, Yehuda Halevy, Joseph Albo, Samson Raphael Hirsch, and Joseph Dov Soloveitchik. This view is expressed by Rabbi Nosson Scherman in the overview to the Artscroll Siddur (p. XIII); note that Scherman goes on to also affirm the Kabbalistic view.
Jewish mysticism ascribes a higher meaning to the purpose of prayer, which is no less than affecting the very fabric of reality itself, restructuring and repairing the universe in a real fashion. In this view, every word of every prayer, and indeed, even every letter of every word, has a precise meaning and a precise effect. Prayers thus literally affect the mystical forces of the universe, and repair the fabric of creation. This approach has been taken by the Chassidei Ashkenaz (German pietists of the Middle-Ages), the Zohar, the Arizal’s Kabbalist tradition, the Ramchal, most of Hassidism, the Vilna Gaon and Jacob Emden.