The music box is a fictional historical artifact that Neal Caffrey was rumored to have in possession during his time as a professional criminal. It is amber in color with four golden cherubs surrounding the top of the music box, one of them being a key to a secret compartment in the box.
It belonged to the Russians as part of the legendary Amber Room, but was stolen by the Nazis during WWII, and considered lost for decades after the war.
Prior to Neal's knowledge of the music box, Vincent Adler heard a rumor that the music box was located in the Amalienborg Palace in Copenhagen. After Alexandra Hunter confirmed the rumor, she partnered with Neal to steal the music box from the palace, but failed.
In "Home Invasion", Alex tracked down the location of the music box to the Italian Consulate in New York. With the addition of Mozzie, the three of them worked together to steal it again, barely succeeding.
Later, it is revealed that the music box carries the equation to a fractal antenna, embedded into its music. In "Under the Radar", Alex tells Neal that her grandfather, Gerhard Wagner, encoded a sunken U-boat's final SOS antenna design into the music box with the intention of finding it later, but he lost the music box during WWII. All he had left was a single golden cherub piece/key to the music box, which he gave to Alex while on his deathbed.
According to Mozzie, the last credible known whereabouts of the music was "Königsberg Castle in Germany in 1945". He also insinuates that the box had been whole until its disappearance.