The "Master" of puppets is a reference to drugs. Throughout the song the "Master" controls you and your life. This is evident in lyrics like, "chop your breakfast on a mirror," "The Master Of Puppets is pulling your strings, twisting your mind and smashing your dreams." Drugs is the Master while the drug user is the puppet. (thanks, Tore - W. Germany)
Metallica recorded this with the San Francisco Symphony in 1999. That version is on their album, S&M.
James Hetfield plays the first solo during the slow instrumental part, Kirk Hammet plays the final, fast heavy solo. While playing the solo, Kirk pulled the top string off of the fretboard of the guitar (usually done by accident when someone bends the high string down instead of up) to make the really high siren-like sound. Everyone loved the way it sounded on the track so they kept it that way.
There are two ways the song is played live. There is one where they just play the song how it is normally played in it's entirety, and another where they play the first 2 verses, and when it's time for the instrumental part they play another song (like "Nothing Else Matters" or "Sanitarium") and when that song is done they continue the final verse of "Master Of Puppets."
This was written by James Hetfield, Lars Ulrich, Kirk Hammett, and Cliff Burton.
At the end of the song, you can hear backwards recordings of the band's guitars while the bandmember's echoed laughter is played. (thanks, Nick - Paramus, NJ, for above 4)
In an MTV Icon special, James Hetfield said that it wasn't until after this song was written that he realized it related to his alcoholism. (thanks, Bertrand - Paris, France)
James Hetfield in Thrasher magazine: "'Master of Puppets' deals pretty much with drugs. How things get switched around, instead of you controlling what you're taking and doing it's drugs controlling you. Like, I went to a party here in San Francisco, there were all these freaks shooting up and geezin' and this other girl was real sick."
When Metallica played two shows in China in 2013, the Chinese government told them not to play this song - perhaps not wanting to harbor unrest with lyrics about being controlled by a greater entity. The band complied, although Kirk Hammett made sure to play the riff during their sets.