Does the period go before or after quotation marks?

Did you ever wonder where does the period go when you are ending your sentence with a quotation? I never had before and that is probably why I have been writing incorrectly my whole life. Let me give an example of what I mean here. Normally I would write the following sentence like this.

After Mike apologized to me I said "don't worry about it".

Notice how my ending period comes after the quotation mark. I have recently learned that this the wrong way to write it. The correct way to write this sentence would be:

After Mike apologized to me I said "don't worry about it."

I have always known that my grammar wasn't perfect but "geez".

Oops, I mean "geez."

I am going to try to make every effort I can now to try and remember to use this proper form of grammar even though it still does not seem right to me. For some reason, I feel whoever made this rule made a terrible mistake. If I made this rule the period would come after the quote, just seems more proper to me.

I also recently learned that this rule of the period coming in front of the ending quotation mark is an American thing. In the United Kingdom, Canada, and islands under the influence of British education, punctuation around quotation marks is normal and accepted as the proper form of grammar. Looks like us Americans got this one wrong! Why did we get it wrong?

"In the days when printing used raised bits of metal, "." and "," were the most delicate, and were in danger of damage (the face of the piece of type might break off from the body, or be bent or dented from above) if they had a '"' on one side and a blank space on the other. Hence the convention arose of always using '."' and ',"' rather than '".' and '",', regardless of logic." This seems to be an argument to return to something more logical, but there is little impetus to do so within the United States.

Now that you know how to end a sentence with a quotation mark, the simple rule to remember is that in the United States, periods and commas (yes, commas too) go inside quotation marks regardless of logic.