I get many visitors searching for the purpose of music, so why not build off of that?
But really, this post is inspired by Chad Orzel's two questions and his conclusion, particularly by a comment made on the religion post.
I pray every day, at least once at night before going to sleep. Most of my prayer is indeed to "understand what [I'm] thinking and feeling," but there is more. I exercise my emotional muscles by reminding myself that I care for those who are sick, poor, or afflicted by injustice. I remind myself that there is much to be thankful for, such as my family, friends, nature, music, good food, good books, and the improvements I have seen in my own life. These daily reminders are a big part of my self-improvement. They don't have to be in the form of a prayer, but I like it that way, knowing that there is someone listening.
I disagree with Rob Knop's distinction between prayer and meditation or reflection. I've been informed by the lovely seminarian living in my house that there are many descriptions of prayer that do not include words. Meditation is a means of feeling the connection with God that Julia described so well. Reflection is a prayer not specifically addressed to God. Washing dishes, running in the woods, or performing music can also be a prayer, either consciously or unconsciously.
So, I pray to be aware of what I value. I pray as a means of strengthening my connections with friends and family. I pray to work out issues I am dealing with. And I pray to explore my relationship with God (Julia is a wordsmith).