# A certain woman named Lydia

"There was a woman named Lydia" 

# a seller of purple

"a merchant who sold purple cloth"

# Thyatira

This is the name of a city.

# worshiped God

A worshiper of God is a Gentile who gives praise to God and follows him, but does not obey all of the Jewish laws.

# The Lord opened her heart to pay attention

"The Lord caused her to listen well and to believe" 

# opened her heart

Here "heart" stands for a person's mind. Also, the author speaks about the "heart" or "mind" as if it were a box that a person could open so it is ready for someone to fill it.