The word Adagio when used in music notation used to mean that the music should be played in a leisurely way or according to the musician's own wishes. Baroque Adagios even left some space for improvisation. Around the 18th century however, it had come to mean just 'slow'.

There are quite a few musical pieces named Adagio, but the most well-known and the one most people mean is Adagio in G minor by Tomaso Albinoni. It has been used and abused over and over again, and you've probably heard it as muzak played in a super market. The Doors used it as musical backdrop in the song The Severed Garden.

The problem is, Albinoni (who lived in the 17th century) didn't actually write the piece himself. It was in fact put together by the 20th century musicologist Remo Giazotto, who based it on material attributed to Albioni.