The console is the text output device for system administration messages. These messages come from the kernel, from the init system and from the system logger.
On modern small computers the console is usually the computer's attached monitor and keyboard.
On many older computers the console is an RS-232 link to a terminal such as a DEC VT100. This terminal is in a locked room and is continually observed by the minicomputer's operators. Large systems from Sun, Hewlett-Packard and IBM still use serial consoles.
It is usually possible to login from the console. A login session from the console is treated by many parts of the operating system as being more trustworthy than a login session from other sources. Logging in as the root super-user from the console is the Command Line of Last Resort when faced with a misbehaving system.