Tefilah is the Hebrew word English translates as "prayer". Recently it has been revealed to me by Rabbi Chava Bahle that this is a poor translation. In English, the word "to pray" means to beg or beseech. The problem is that tefilah does not mean that. Rather it means to 1) self-reflect and 2) taking a wide range of things and unifying them.
The point being that prayer is a tool God uses to change us rather than a tool we use to change God.
Prayer is also the practice of being able to step back and reflect on how it is that contradictory things are actually unified in some way. Being able to "see" the unity in the midst of a broken world is very important. The genius and beauty of the Lord's Prayer is Jesus' ability to take a wide range of things (thankfulness, the greatest commandment, hope, dream, praise, etc.) and put them all together. Additionally the prayer takes things that seem contradictory and unites them, such as praying that heaven will come on earth.
Prayer changes our hearts and helps us see. This is why those who pray know the power it has to change us. Perhaps that is why many of us do not pray - at some level we know it will change us and we fear that change.