Newton's First Law states that an object will remain at rest or in motion in a straight line unless acted upon by an external force. It may be seen as a statement about inertia-objects will remain in their state of motion unless a force acts to change the motion.
When a dancer prepares to spin, she must plie, put out the arm that will propel her in the desired direction, and then push off into passe. The external force in this action would be the dancer's foot as she pushes herself off the ground to spin.
Newton's second law of motion states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
When a dancer is mid-turn, she will spin as hard and as fast as she pushes herself off the ground. The more force that she puts into her plie when she prepares to spin, the faster and longer she will spin.
Newton's third law states that for every action, there is an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object.
While the dancer is spinning, there is an opposite force pushing against her and trying to pull her in the opposite direction, which is why you have to push off with your outer arm facing the way you want to go.