What do the red numbers at the main menu mean?

This is a list of menu items where a red number can occasionally appear, including the numbers description:

• Main page - a number of games where it is your turn to move. Click on the number to play the first of them.
• Waiting games - a number of open game invitations (you can accept them).
• Tournaments - a number of open tournaments (you can sign up for them).
• Message box - a number of new (unread) messages waiting for you.