Thanks for providing a screenshot, which is always helpful. It shows the People tab on an Organisation record.  This means that you're looking at a list of relationships that have been created between ABC Company and People on your database.  It looks very much like John Smith has been related to ABC Company twice, which is different from there being two People records for John Smith.

Sometimes it's appropriate to create more than one relationship between records on your database.  For example, your Organisation might be both a 'supplier to' and a 'customer of' another Organisation or, returning to this example, John Smith might be an 'employee of' ABC Organisation AND be a 'partner of' or 'contractor to', etc.

I suspect that when you've tried to delete what looked like a duplicate record, you opened the Person record and deleted that which would explain why it removes both rows from the view you see in the People tab on the Organisation record.

Firstly, I'd recommend activating the Relationship column on the People tab, which will help you see which relationships have been created.  If different types of relationship have been added you might want to leave the record as it is.  However, if you do want to delete one of the relationships, scroll to the far right until you can see the delete button.  If you hover over this you'll see infotext that it says this will remove the relationship (as opposed to deleting the Person record), thus deleting one relationship will leave the other one in place.

Click on the screenshot below to enlarge.

Hopefully that sorts out the issue for you but if not, please let us know.