A whisper is a chat message sent directly to another user. When you whisper, the recipient is the only person who can hear what you say.
To whisper to someone, first select their name in the "Whisper To" list. Then simply type your message in the whisper field and press Enter like you would for a normal chat message. You can also whisper to someone by selecting Whisper from the avatar pop-up menu.
Whispers appear in your chat history window in blue italics to distinguish them form normal chat. Whispers that you send are listed along with the name of the person you whispered to.
Note that you can quickly switch back and forth between the normal chat field, the whisper list and the whisper chat field by simply pressing the Tab key.
Also note that in order to whisper to someone, they must be physically nearby and they must have already revealed their identity by speaking (either via normal chat, or by first whispering to you.) Mute blocks whispers as well as normal chat. If there is currently no one around to whisper to, the whisper field will automatically disable itself.
Be aware that in some worlds the ability to whisper may be disabled for tourists. This option is set in the world options.
If you don't want to whisper you can hide the extra whisper controls from the Show Menu or by pressing the F8 key.