An Earthcache is a special type of cache which does not have a container or logbook. The intention of these caches are to take you to a geologically significant or unique location and show and teach you about Earth Science.
Earthcaches require that you answer a few questions regarding the location and optionally take a photo of yourself or your GPSr at the location. You send your answers to the Earthcache owner and they let you know if you answered correctly.
Many Earthcache owners write in the description that you must wait for their approval before logging your find but according to Groundspeak Geocaching guidelines this is incorrect. You can send the answers to the Earthcache owner and log the find. If you answered incorrectly the Earthcache owner can delete your find log.
1. People do not need to wait for permission to log your EarthCache. Requiring someone to wait is not supported by the EarthCache guidelines. People should send their logging task answers to you, then log your EarthCache. When you review their logging task answers, if there is a problem, you should contact them to resolve it. If there is no problem, then their log simply stands.
Some of the Frequently Asked Questions (FAQ) can be found HERE at the official Earthcache.org website.
There are much fewer Earthcaches then Tradition, Multi, or Mystery caches but most Earthcaches are much more interesting then the container type caches. If you have not done one I would recommend you make it a point to find one near you and give it a try.