A whisper, also called a "tell" is a message sent to a specific player. No one else can read a whisper but the player it was sent to. Whispers can be sent to anyone on the same realm, it's not limited to both players being in the same zone or group. A player must be online to receive your whisper. If you try to whisper a player that isn't online you will get a message stating that the player cannot be found.
To whisper another player, type /w playername in your chat window (replacing playername with the name of the character that you want to whisper). As soon as you hit the space bar after the player's name, the chat prompt will change to indicate that you're whispering someone. You can also click on a player's name in the chat window and that will automatically open a whisper prompt for you.
Whispers are a way for you to communicate with your friends in the game without having to be in the same group or zone. They can also be helpful in negotiating a sale with someone advertising in the Trade Channel in a major city. For example, if someone in Trade says: "Selling 4 stacks heavy leather, PST" It means they want you to whisper them to discuss price and delivery method.
It's considered polite to use whispers to talk to someone before inviting them to a group. While in a group, if you have constructive criticism for a fellow group member it will often be better received if you have that conversation using whispers/tells than if you call them out in party chat.