No all the stars in the sky doesn't move.
The Earth rotates from West to East on its own axis. Hence, all stars in the sky (except the Pole star) appear to move from East to West.
With reference to the Earth, the Pole star does not appear to move in the sky because it is located above the axis of rotation of the Earth i.e., in the north direction. Thus it appears to remain stationary at a point in the sky.