Philemon was a well-to-do Christian of Colossae. He and Paul were evidently good friends.
Onesimus, a a slave who belonged to Philemon, stole some money from Philemon and ran away to Rome. While in Rome, Onesimus either ran into Paul or looked him up. At some point after getting together with Paul, Onesimus became a Christian.
Paul then took the initiative to intercede on Onesimus' part by writing a letter to Philemon, beseeching him to forgive his runaway slave. Paul himself offered to repay the stolen money.
We do not know how Philemon received Paul's letter, but it is rumored that Onesimus afterward became a church bishop in Berea.