Yes, the stars appear to move in the sky. However, this apparent motion is not due to the actual movement of the stars themselves, but rather caused by the rotation of the Earth on its axis.
The Earth rotates from west to east, completing one full rotation in approximately 24 hours, causing the celestial sphere (an imaginary sphere surrounding the Earth) to appear to rotate as well. This rotation gives the illusion that the stars are moving across the sky.
The movement of the stars can be observed over the course of a night. As the Earth rotates, different stars come into view while others disappear below the horizon. The stars in the northern hemisphere appear to revolve around the North Star (Polaris), which remains relatively stationary due to its alignment with the Earth's axis of rotation. In the southern hemisphere, there is no single star that serves as a direct counterpart to Polaris, but the southern celestial pole is close to the faint star Sigma Octantis.
It's important to note that while the stars appear to move relative to our vantage point on Earth, they are actually fixed in their own positions in space. The apparent motion is a result of the Earth's rotation, and the vast distances between stars make their movements negligible over human timescales.