The Naval Observatory was created for the purpose of doing astronomy and astronomical measurements that are vital for the Navy. This includes the positions of stars (for navigation), the exact time, and positions of the sun, moon and planets. Their mission statement outlines this, though I find the second part of point four to be darkly humorous. It reminds us that, although the USNO hires astronomers, it is still a branch of the military.
Most of the people at the Flagstaff station work on the positions, distances, and motions of stars. Yes, stars move, albeit usually much too slowly for people to detect. The "fastest" moving stars still take a thousand years to move a distance the size of the full moon through the sky. Still, these motions are important for understanding the structure of the Milky Way galaxy.
My thanks to the staff at the USNO for being such gracious hosts!