Full answer
The Moon orbits Earth in an ellipse, not a perfect circle, so the distance changes constantly. The closest point (perigee) is about 225,623 miles (363,300 km) and the farthest point (apogee) is about 252,088 miles (405,500 km).
The widely cited average distance of 238,855 miles (384,400 km) is the mean of these extremes. Light covers this distance in about 1.3 seconds, which is why there's a slight delay in communications between Earth and lunar missions.
Astronomers measure the Moon's exact distance using laser ranging — NASA's Apollo missions left retroreflectors on the lunar surface, and observatories bounce laser pulses off them to measure round-trip travel time with millimeter precision.
The Moon is also very gradually moving away from Earth at about 1.5 inches (3.8 cm) per year, driven by tidal interactions. Billions of years ago it was much closer and would have appeared far larger in the sky.