This is an excellent question. Many people are unsure how to use it appropriately. I'm not an English expert in any way shape or form, but I'll share what I do know.
In British English, it follows the more logical method...
My Mom says I am a "slob", but I think it is unfair to call me a "slob".
On the other hand, American English always requires periods and commas to go inside the quotation marks, irregardless of what would be linguistically logical.
My Mom says I am a "slob," but I think it is unfair to call me a "slob."
The only real exception I'm aware of is regarding quoted words or statements that flow in your sentence without the need for a break or pause.
The phrase "chivalry is dead" suggests that the expectations of men in today's society has changed.
In these cases, you bypass the rule by avoiding the usage of a comma/period.
If there is anyone who is more experienced than I, please share your feedback!