1-7: This letter was written to a Christian named Philemon, who hosted a house church. The focus of Paul's letter was Philemon's slave, Onisimus, who stole money from Philemon and ran away to Rome. While in Rome, Onisimus met the imprisoned Paul through circumstances not recorded in scripture. Paul shared the Gospel with Onesimus, who responded in repentance and trusted in Jesus.
8-14: Paul sent Onesimus back to Philemon with this letter, which explains Onesimus' conversion to Christianity. The "father/child" relationship Paul had with Onesimus was a spiritual relationship: meaning Paul lead him to the Lord and discipled him. Onesimus was very useful to Paul while in prison, yet he still sent him back to his master, Philemon. Paul wanted Philemon's generosity to be voluntary regarding his decision of wether or not to let Onesimus stay in Rome, so he didn't command that Onesimus should stay.
15-16: Paul suggests that God sovereignly guided this whole scenario so Oneisimus would be saved.
17-21: Paul modeled the Gospel by paying Onesimus' debt to Philemon, making peace between them.
22: He was confident Philemon's prayers would be answered and he'd be released from prison, so Paul asked for him to prepare a room for him.
23-25: The letter is closed with greetings from Paul's companions.