Whisper is one method of chatting privately in game. You can choose who you wish to whisper to by pressing " [Shift-apostrophe], typing in the name of the character you wish to talk to, and pressing enter. You may then enter your message or press Esc to cancel. Note that you will by default whisper whoever last whispered you, and you can also select previous characters who have whispered you by pressing the up arrow.
Whispers can reach anyone who is logged into the game, with a few exceptions. Those who have put you on their Enemies list will ignore you, as will those who have turned off their ability to hear whispers. Also, in some areas of the game (such as the Revels judging room) prevent whispers from reaching those inside.
If you whisper someone who is reading a board, they will automatically respond with a message to that effect. However, the character will still hear your whisper, and may respond later.