There are two options for messaging people on the site.
1. Use the in-game "message opponent" button, which allows you to send a message with a move whenever it is your turn to move. These messages are part of the "message log" on the game board.
2. Use the messages system, which is like on-site email, and send a private message to anyone (opponent or not) at any time. These messages are kept in a different place, the separate "MyMessages" system, which you can access via your profile button (top right hand corner of page). Your opponent's screen will light up the middle "messages" icon when they receive your message.