The morgue is a folder of morgue files (character dumps); .txt files that give a summation of everything that happened during your game. It stores both current information, like inventory and skills, and a history of actions, levels, and notes. The rcfile gives more options as to what is and isn't logged as a note. Sharing a morgue after a particularly gruesome death is both an informative and entertaining thing to do, though you can generate a morgue file for living characters, too.
You can update your morgue file by pressing #. In addition, the # command will either print a link to your morgue (online) or the directory to your morgue folder (offline).
Morgue files are not to be confused with a bones file, which is used to generate player ghosts.