"One of those listening was a woman named Lydia, a dealer in purple cloth from the city of Thyatira, who was a worshiper of God. The Lord opened her heart to respond to Paul's message." NIV translation
This too can be a somewhat puzzling verse to read as we are told that Lydia was a "worshiper of God" but then we are told that "the Lord opened her heart". You see, many people want to worship God but that does not change the fact that, to have a right relationship with God, we must accept his Son, Jesus, as the payment for our sins.
"When she and the members of her household were baptized, she invited us to her home. 'If you consider me a believer in the Lord,' she said, 'come and stay at my house.' And she persuaded us." NIV translation
This verse affirms a pattern that occurs throughout the entire Bible. That is the pattern of people being baptized when they come to Christ. When Lydia accepted the message of Jesus Christ, we see that her entire household was baptized. Likewise, we see the pattern of hospitality in that, as a Christian, she offered her home to Paul and his companions.
"Once when we were going to the place of prayer, we were met by a female slave who had a spirit by which she predicted the future. She earned a great deal of money for her owners by fortune-telling." NIV translation
So, one day as they were heading to the river to meet the other worshipers, they were met by a woman that we would call a "psychic" today. As we see, this woman was a slave to the spirit as well as being used by others for their own profit.
"She followed Paul and the rest of us, shouting, 'These men are servants of the Most High God, who are telling you the way to be saved.' She kept this up for many days. Finally Paul became so annoyed that he turned around and said to the spirit, 'In the name of Jesus Christ I command you to come out of her!' At that moment the spirit left her." NIV translation
This might strike us as strange that Paul would drive the spirit out as it was saying what appeared to be good things. Nevertheless, Paul recognized the fact that the spirit was demonic and he cast it out of her. Today, we have become accustomed to seeing these "pschics" and we forget the fact that they are actually filled with a demonic spirit.