* Period: Use a period inside the closing quotation marks if the dialogue is a complete sentence and stands alone.
*Example:* "I'm going to the store," she said.
* Comma: Use a comma inside the closing quotation marks if the dialogue is not a complete sentence, or if it's part of a larger sentence.
*Example:* She said, "I'm going to the store," and then left. (The entire sentence is "She said, 'I'm going to the store,' and then left.")
*Example:* "I'm going," she said quickly, "but I'll be back soon." (Each quote is part of a larger sentence).
In short: If the quote could stand alone as a complete sentence, use a period. If it's grammatically incomplete on its own, use a comma. Always place the punctuation *inside* the closing quotation marks (in American English style).