At camp Full Moon, the Snatcher is said to take one of the campers every year. In one of the camp fire stories, it is said that it snatched a camper named Johnny.
When Dustin goes to Camp Full Moon, he doesn't believe the legend at first, until he discovers that Laura is the Snatcher. He discovers this when she transforms into a fox, and attacks him. However, when a zombie-like hand drags her into the river and tears her apart, the Snatcher is destroyed.
The Snatcher can take the form of whatever it would like, such as a fox or a spirit. Throughout the book, it was in the form of a girl named Laura.