Prayer is simply communication with God, and is important much like communication between a husband and a wife is important. Part of it is the act of releasing our stresses and our burdens onto him and part of it is thanking him and admitting our faults to him despite the fact the he already knows these things. Much like a husband may know that his wife struggles with a certain habit, her confession of that to her spouse greatly affects both parties, because now they can work together to resolve the issue. In addition, though my wife knows she got me exactly what I wanted for my birthday, my act of expressing my gratitude to her brings her great joy and draws us closer together.
There are too points that must be considered about the purpose of prayer. The Bible tells us that God knows what we need before we ever ask him (Psalm 139:4; Matthew 6:8) but also that the prayers of a righteous man avail much (James 5:16). How we as Christians can take this into practice is that while God knows our needs and wants, our action of bringing them before him is our way of putting our hearts and minds in line and submission to his will. As a result us making our requests known to God (Philippians 4:6), he will respond to us according to his will. while his answer to us may be yes, we need also except that his answer can also be no, and other times it can also be later.
Prayer groups serve a similar purpose in that instead of one person coming before God, it is a group of people coming before God with one (or multiple) person as the spokesman. By this the group can hold each other accountable to promises that are made in the prayer, things that the group needs to repent of, and joys that the group can experience together. So for example, at a church service when the pastor says the closing prayer and the congregation says “amen,” what they are saying is “we agree with what is being said, we agree that this is where we need work and we ask for your help in these areas.”