The musical jug is precisely what it says it is: usually a stoneware jug (although glass and ceramic jugs are also played) into which the player blows with their mouth. The musical jug is played in a manner similar to playing brass instruments or didjeridoos. Although, whereas brass instruments or didjeridoos are held directly up to the mouth, the musical jug is held about an inch from the mouth as the player buzzes into it. Pitch is changed by altering the shape of the embouchure and tightening or loosening the lips.
The jug band trend became popular during the 1920s, and grew out of the jazz and country blues traditions. Here's a tutorial on how to play the musical jug.