A nickname is a shortcut to a person's email address. If you enter someone's email address, and you enter a shorter name for that person in the Nickname section of that person's contact information, you can use the nickname associated with that person to send email.
For example, you can enter "Joe" as the nickname for josephwilliams@someplace.com. Then, when you are composing an email using the Webmail interface, you don't have to enter the entire email address. Just enter the name "joe," and the email will be sent to the address josephwilliams@someplace.com.