How does astronomy benefit society?

Many developments in optics come from astronomy. The Galilean refracting telescope and the Newtonian reflecting telescope form the basis of pretty much every optical system in the world, including disposable cameras, photocopy machines, TV cameras and more things than I can list. Modern astronomy continues to push the limits of optical systems, so advances end up in things like weather satellites. Astronomy also pushes detector technology, with similar results. Earth observation satellites also need to know where they are pointing, and they figure that out by looking at the stars. Astronomical observations provide verification of theories of basic physics, so an understanding of our immediate world is enhanced by observing far-off galaxies. The above answer is well presented, so I will take a completely different tack. Hundreds and thousands of years ago, people, kings, and religious leaders believed our mortal existence was linked somehow to the heavens. An eclipse, or the sighting of a comet, were often regarded as heavenly events that had certain significance upon those who inhabited Earth. Even the story of the birth of Christ, and many other religious events, are linked to celestial landmarks. People turned to astrologers to help explain and predict events on earth. For some part, astrologers relied on historical data to unravel the cosmic mysteries, but a great deal of astrology was borne out of hearsay, intuition, and stories. During the renaissance period, when art and science enjoyed a revival in the Western world, men and women of scientific reason began to examine the heavens in a more critical and objective way. While many of these people were ostracized and punished (the most famous, but by no means to most gruesome, was Galileo, who was severely sanctioned by the Church for turning his telescope skyward and reported what he saw), eventually there became an social awakening through which people viewed their world, and space, with a more open and inquisitive regard. While astronomy is by no means the only field of science in which this awakening took place, astronomy in particular delivered a profound blow on the monopoly the Church held upon public thought. The heavens were, after all, the exclusive dominion of the Church. Once people accepted that celestial phenomenon could be better explained through scientific study than through superstitious beliefs, all peoples benefited through an enlightened and progressive understanding of the universe that we live in. At its simplest, without an understanding of planetary orbital dynamics, we would not have communication satellites, global positioning systems, or the weather forecasting we take for granted. Without astronomy we would not understand the damage our carbon-burning industries inflict upon the ozone layer or the rain forests. We would not understand how solar flares affect terrestrial communications, or how the El Nino and La Nina impact our global climates. On the more esoteric side, through astronomy and astrophysics, we have a comprehensive awareness of the universe's origins, of how our own solar system began, and what lies in waiting for our own planet in the millions of years to come. We understand the perils of global catastrophic events, such as the asteroid that wiped out 90% of all life (including the dinosaurs) 65 million years ago, and we are formulating the first inklings of how to ward off further occurrences of such events. Finally, through space exploration - in person, via probes, or through our advanced telescopes - we have reached out to other worlds - men on the moon - the NASA rovers on Mars - even Voyager now streams beyond our earthly bonds in search of wonders elsewhere. Within the past 10 years, we have detected thousands of far-away planets - and a few remarkably similar to Earth. Imagine someday detecting a planet inhabited by advanced life - knowing we are no longer alone in the universe. Like all sciences, astronomy and astrophysics have served to enrich our understanding of ourselves and our environs, and have led to discoveries and inventions (of all stripes) that have helped and hindered our short stay here on Earth.