The related checkbox shows everything related to a record once it is checked. It is useful if you have one contact and many jobs on it and you are searching for a particular record. However, if you have it checked it may show unnecessary or information that is not relevant to the contact. Usually, if you uncheck it in the activity, tasks, attachments, financials, work orders tabs you can solve that issue.
Payments always show whether the related checkbox is clicked or not. A payment will show on a contact and in the job area. If you want to see where a financial record was recorded uncheck the related box on the estimates/invoices. If the estimate/invoice disappears it was not created on that record. At times an employee may accidentally record an invoice or estimate under the contact and not the particular job. This isn't necessarily that big of a deal because it will still sync to QuickBooks if you have QuickBooks but it is not as organized as it should be. You could leave the record as is or delete it and re-create a new one in the correct location.
When checked, it pulls all records from all related records to the list.
So if the customer saw invoices on a list and then unchecked the related checkbox, they would no longer be visible there if they actually belong to a different record like a job.