If someone was fired from a retail store and then applied to the same store at a different location will the employer find out?

Of course. Computers are used for payroll records, right?

Answer

It depends. Sometimes different branches of the same business are owned by different companies. For example, two McDonald's or two Hampton Inn Hotels will not have any info about each other's payroll records. However, I would be tempted to put the previous job on your application if it's good experience, and just list "personal" under reason for leaving.