It depends. There are differences between American and British convention; I'm British, so I use these rules:
If the quotation marks enclose a word or a short phrase, the full stop/period usually goes outside of the quotation marks:
Queen Victoria said they were "not amused".
If the quotation marks enclose an entire sentence, the full stop/period goes inside the quotation marks:
Queen Victoria said, "We are not amused."
In the first example, we are only quoting a fragment, and this fragment doesn't form a complete sentence on its own; therefore the period doesn't belong to the quoted fragment.
In the second example, we are quoting an entire sentence which begins with a capital letter and ends with a period, so the period belongs to the quotation itself, as well as the longer sentence.