A coil is a spiral shape or a series of circles, each one inside another. You might see a coil of rope on a dock or coils of hair in a fancy hairdo.
A snake naturally curls its body into a coil, and the end of a whip after it's been cracked tends to form a coil. To move in this way is also to coil, as when your cat coils around your leg or you coil a necklace around your finger. This verb form is the earliest meaning of coil, from the Middle French coillir, "to gather," with the Latin root colligere, "to gather together."