This depends which side of the pond you're on. If I understand correctly, British speakers usually put the punctuation outside, but there are exceptions. Americans, on the other hand, are taught to put periods (that's "full stops" if you're British) and commas inside the quotation mark.
This dates back to the days of mechanical printing using hand-set movable type. Supposedly, if a perod or comma were put outside a quote mark, it might break off.
Another quaint convention that dates back to those days is the double space after a sentence-ending period. This makes no sense with modern computerized fonts, but it is still a proofing option in word processing; e.g. Microsoft Word.