International Observe the Moon Night is a worldwide celebration of lunar science and exploration held annually since 2010. One day each year, everyone on Earth is invited to observe and learn about the Moon together, and to celebrate the cultural and personal connections we all have with our nearest neighbor.
Amateur and professional astronomers, at countless locations across the world, will be sharing views of the Moon with their communities; the International Observe the Moon Night website has list of scheduled events. You can register your event here, and find resources for hosting your event here.
The NASA Night Sky Network (NSN) also has events listed in the U.S. being hosted by astronomy clubs, museums, libraries, and volunteer NASA/JPL Solar System Ambassadors. The NSN is a great way to find astronomy clubs and events near you (in the U.S.).
The event occurs in September or October, when the Moon is around first quarter. A first quarter Moon is visible in the afternoon and evening, a convenient time for most hosts and participants. Furthermore, the best lunar observing is typically along the dusk/dawn terminator, where shadows are the longest, rather than at full Moon.