1. The operator's station of a mainframe. In times past, this was a privileged location that conveyed godlike powers to anyone with fingers on its keys. Under Unix and other modern timesharing OSes, such privileges are guarded by passwords instead, and the console is just the tty the system was booted from. Some of the mystique remains, however, and it is traditional for sysadmins to post urgent messages to all users from the console (on Unix, /dev/console).
2. On microcomputer Unix boxes, the main screen and keyboard (as opposed to character-only terminals talking to a serial port). Typically only the console can do real graphics or run X.