console

/dev/tty        Current TTY device
/dev/console    System console (set as a parameter at boot time)
/dev/tty0       Current virtual console
/dev/tty1       Ctrl+Alt+F1

*  controlling terminal for a session, like sending signals (Ctrl+C)
*

console:

grub.conf
# default points to /dev/tty0
console=ttyS0
After that your /dev/tty0 is a monitor and /dev/console is /dev/ttyS0.

tty0

a always current console.

tty :

the console used by the process querying it.
# the console where process starts
Unlike the other devices, you do not need root privileges to write to it.