There are three semicircular canals filled with fluid in the inner ear at right angles to each other. Each senses the movement of your head along a different axis. Using all three, the brain is able to orient your body in 3D space. Tiny hairs lining the canals (hair cells) sense motion by sloshing of the water in the canals. Information from the hair cells are relayed to the brain as an interpretation of movement of the head.
During spinning in a circle, the fluid in the canals initially slosh in the direction opposite to your head's motion. However, the jelly-like material in which the hair cells are imbedded, the endolymph, adjusts to the movement of your head,causing the hair cells to straighten, and your brain no longer receives the message that you're spinning. In effect, your brain thinks that you are standing still while the the world is rotating around you.
When you suddenly stop spinning, because of inertia, the endolymph keeps spinning. The fluid in your ear moves again in the direction in which you were spinning before, and a signal of movement is transmitted to the brain. You sense that you are moving, but you're not.
