That sounds like application level security (something you code into your web page or program) rather than database security (something you can set/manage via the database engine). Typically, all of the users will use a single database login, so the database doesn't "know" which user is connected, so you really can't manage row level security at that level... It needs to be done by the application.
Do you have any resources where you could point me where I could learn about this topic?
After the user logs in (and is checked against the registration database), I want PHP to prevent them from altering any records but their 'own'. If their login info was saved as a session/cookie, could the program 'know' which records they should have access to?