The original Toy Story won one Academy Award.
At the 68th Academy Awards, the film received a Special Achievement Award honoring director John Lasseter for the development and inspired application of techniques that made the first feature-length computer-animated film possible. That honorary-style recognition counts as one Academy Award win for the film.
Toy Story was also nominated in three categories: Best Original Screenplay, Best Original Song for “You’ve Got a Friend in Me,” and Best Original Score. It did not win those competitive nominations, so the film’s total is one Oscar rather than three or four.
Another source of confusion is the timing of the Best Animated Feature category. That category did not yet exist when Toy Story competed at the 1996 ceremony; it was introduced later, beginning with the 74th Academy Awards for films released in 2001. Toy Story’s single win therefore combines historic technical achievement with the award system of its era.