Paul Morphy and a Queen Sacrifice

I am still working my way through the book 500 Master Games of Chess and I am up to chapter 7 which is called Four Knights and Three Knights Game. The first game I came across is between Louis Paulsen and Paul Morphy in New York in 1857.

There is a small paragraph before going through the game which says "Although this game is adorned by one of the most famous Queen sacrifices in the literature of the game..." Well, that had me! My queen usually dies in the oh-so-common way of QxQ, but I want to learn how to sacrifice pieces properly.

The paragraph also went on to say that this is the game that spread Morphy's fame to Europe. 

While many people love this game (me included) the computer here on Chess dot com was not so impressed. Morphy had two different times he could have checkmated Paulsen at an earlier stage of the game. The computer also pointed out that Paulsen also had a chance at checkmate!

Here is the game with the computer's suggestions: