Peace means reconciliation
Peace with God is more than a quiet feeling after a difficult day. In the Bible, sin puts people at odds with the holy God who made them. Paul even describes people as enemies before they are reconciled to God through the death of his Son (Romans 5:10).
Jesus deals with that broken relationship. His death was not simply an inspiring example of love. Through Christ, God makes peace and brings people back to himself (Colossians 1:20). So peace with God means the hostility caused by sin has been answered by God's mercy.
Therefore being justified by faith, we have peace with God through our Lord Jesus Christ:
It is received by faith
Paul says this peace comes through being justified by faith. To be justified means that God declares a person right in his sight, not because they have lived perfectly, but because they rely on Christ. It is a gift of grace, not a reward for getting your life together first (Romans 3:23-24).
That doesn't mean sin is unimportant. It means Jesus is sufficient. God showed his love by sending Christ to die for people while they were still sinners (Romans 5:8). Faith is trusting him with what you cannot repair on your own.
But God commendeth his love toward us, in that, while we were yet sinners, Christ died for us.
It is not the same as always feeling peaceful
A Christian can have peace with God and still feel anxious, grieve deeply, or face a painful season. Peace with God describes where you stand with him because of Christ. It does not promise that every emotion will immediately settle.
Yet God also invites his people to bring their worries to him. Paul tells the Philippians that God's peace guards the hearts and minds of those who pray with thanksgiving (Philippians 4:6-7). That is often a lived, repeated experience of trusting God in the middle of trouble.
Be careful for nothing; but in every thing by prayer and supplication with thanksgiving let your requests be made known unto God. And the peace of God, which passeth all understanding, shall keep your hearts and minds through Christ Jesus.
Living from peace with God
If you have trusted Christ, you do not need to approach God as someone hoping to persuade a reluctant judge. Through Jesus, you have access into grace and can come to God honestly, with both gratitude and need (Romans 5:2). Prayer, repentance, and obedience then become responses to God's welcome, not attempts to purchase it.
If you are unsure whether you have this peace, start simply. Read the accounts of Jesus, tell God where you are, and place your trust in him. Jesus says that those who come to him will not be cast out (John 6:37).