How to ensure a user can only delete it's records?

Yep! Authorization (gates/policies). 2 different ways, depending on what you want to do/how you want to organize them. https://laravel.com/docs/5.6/authorization

Perfect! Thanks!

Depends on what restrictions you want to have for your users :) If it is something really really simple, you could add a "role" column to your User table and make a middleware to check if the user is authorized to perform an action. Otherwise go for gates/policies. Or use both :)

Correct; my goal is to ensure a user can edit or delete records assign to him and to make sure one user can't delete records for another user.