The very center of a shooting or archery target is a bull's eye. An archer gets ten points if her arrow hits the bull's eye.
The familiar design of ever-smaller circles within circles is a bull's eye, whether it's an actual target or just resembles one. Darts players might yell, "Bull's eye!" when they hit that center ring. The term originated with 1880s shooting competitions in England, either from the idea that the black circle in the middle of the target looks like the eye of a bull, or possibly a five shilling coin that was popularly called a bull's eye.