Family background refers to your birth environment. For example, some people come from official families, some from rich families, and some from poor families. These are considered to be a person's family background. Education background refers to the environment in which you receive education, such as whether your middle school or university is a famous school and whether the teachers are well-known teachers and professors.
As for what is a person with a story, this is rather general, but it is generally accepted that if you have done something that others think is important in your life, such as doing good deeds, finding money and not knowing what to do, or being rewarded and punished severely, such as being an excellent citizen, an excellent model, having been in prison, or having made major inventions and discoveries, it counts.