The most basic definition of prayer is "talking to God." Anyone can do it. In your mind, you formulate something you want to say to God, then you say it. People pray all the time, all around the world, to a variety of deities. Indeed, prayer forms the basic building block of spirituality.
But the Bible has a bit of a different take on prayer. It still involves talking to God, and anyone can still do it. A biblical approach to prayer, though, involves responding to God's revelation. The author of Hebrews writes that God has, at various times and in diverse ways, revealed himself through the prophets. But now, Hebrews says, God has revealed himself in the person of Jesus Christ (Hebrews 1:1-2). The whole Bible—the Hebrew Bible and the New Testament—is the perfect chronicle of God's self-revelation.
The truth is that there is one God, the Father Almighty, who created heaven and earth; that there is one God who exists in the persons of the Father, the Son, and the Holy Spirit; and that this God is the only worthy recipient of worship. Biblical spirituality involves hearing or reading God's Word (the Bible), and responding in prayer.
When we pray, what are we responding to, exactly? Well, the Bible is a diverse library, of 66 individual books with literary genres consisting of poetry, prophecy, history, genealogy, and sermons. But all of these books and genres contain one overarching story: the creation of the world, the fall of humankind, God's redemption of humankind, and God's reconciling of the world to himself. When you read the Bible, you will see how God's plan is astonishing and unified. You will see, among other things, what Israel's exodus from Egypt has to do with Jesus' invitation for you to repent of your sins.
So what is prayer? Prayer is responding to God's call to you to repent of your sins and join Him in His plan for the world. In the Bible you see God for who He is, and you see yourself for who you are. This reality moves Christians to cry out with Isaiah, "Woe is me! For I am lost; for I am a man of unclean lips, and I dwell in the midst of a people of unclean lips; for my eyes have seen the King, the LORD of hosts!" (Isaiah 6:5) After that Christians are assured of God's forgiveness, because "If we confess our sins, He is faithful and just to forgive us our sins and to cleanse us from all unrighteousness" (1 John 1:9).
From there we move to giving thanks to God for various gifts he's given us, or bringing our requests to Him. When we pray we can ask God for help, or we can even express our lack of understanding about aspects of the life He has called us to.
The Lord is near to all who call on Him (Psalm 145:18).