What is the practical application of the story of the woman caught in adultery? John 8:2-11 tells us the story of the woman caught in adultery:

2At dawn [Jesus] appeared again in the temple courts, where all the people gathered around him, and he sat down to teach them. 3The teachers of the law and the Pharisees brought in a woman caught in adultery. They made her stand before the group 4and said to Jesus, “Teacher, this woman was caught in the act of adultery. 5In the Law Moses commanded us to stone such women. Now what do you say?” 6They were using this question as a trap, in order to have a basis for accusing him.
But Jesus bent down and started to write on the ground with his finger. 7When they kept on questioning him, he straightened up and said to them, “Let any one of you who is without sin be the first to throw a stone at her.” 8Again he stooped down and wrote on the ground.
9At this, those who heard began to go away one at a time, the older ones first, until only Jesus was left, with the woman still standing there. 10Jesus straightened up and asked her, “Woman, where are they? Has no one condemned you?”
11“No one, sir,” she said.
“Then neither do I condemn you,” Jesus declared. “Go now and leave your life of sin.”

It's a wonderful story of Christ's compassion on an obvious sinner... as well as a story of Christ's cleverness when dealing with the Pharisees.
But what is the practical application of this story?
In the story, everyone there recognised that they were sinners, including the top religious leaders. We should recognise that in ourselves also, no matter how 'good' we try to be. Jesus' compassion is not just for the woman but also for all of us.

This comes to the heart of Christianity and is the whole point of John's Gospel: everyone has sinned/rebelled against God and faces his just judgement. But God sent his Son Jesus to come and save such sinners, and we're invited to trust in him so that we're no longer condemned by God.