Disney had been practically printing money since making Snow White and the Seven Dwarves—its first animated film—way back in 1937. But, by the 1980's computers had started to become powerful enough to create the same high-quality product as traditional animation. Move over hand-drawn princesses, 3D computer animation had come to town.
In 1995, Toy Story had the distinction of being the first ever full-length computer-animated film. Sure, some movies had used computer-animation for select parts—the ballroom scene in Beauty and the Beast and the escape from the Cave of Wonders in Aladdin are two examples—but, Toy Story took what computers could achieve to a whole new level.
And it forced traditional animators to break open their laptops and learn some new skills.
The difference between 2D hand-drawn animation and 3D computer-generated animation is pretty much how the finished product looks. Traditional animation is done by hand and keeps its flat look on screen. Computer animation gives characters more of a three-dimensional feel and can be done on just one computer if needed.
The Lion King, which came out in 1994, had a staff of 800 while Toy Story was done with just 110 people. You know film companies just love having to pay less people while still making boatloads of cash. (Source)
Today, tons of cartoons (in fact, most feature-length cartoons) feature computer animation. Shrek, Ice Age, The Polar Express, and Madagascar all owe a debt of gratitude to their big brother, Toy Story.