Billy is a puppet that has appeared in the Saw franchise. It was used by John Kramer to communicate with his test subjects by delivering recorded messages, often appearing on a television screen or occasionally in person to describe the details of the traps and the means by which the test subjects could survive.
In the film series, before becoming the Jigsaw Killer, Kramer was believed to have created Billy to be given as a toy to his unborn child. However, Jill, his wife, is shown to miscarry after being hit in the stomach by a swinging hospital door due to Cecil ramming it open without looking. The unresolved anger caused to Kramer by this loss may have been a motive for making the puppet a vehicle for his instructions to his victims in the traps. The traps he created can be seen as a manifestation of his anger with people who he feels don't appreciate having their life when his unborn baby was killed before having a life of its own.
Although it has never actually been identified in the films, "Billy" is the name by which it is referred to by writers, directors, and members of the cast and crew on documentaries and interviews. The name was given to it by its real-life creator James Wan, who is the director and co-writer of Saw. There is disagreement that the name is spelled "Billie", although Wan himself spells it "Billy".