When you switch on private messaging at Admin Panel -> Settings -> Switches, members will be able to contact each other on your site without email. You frequently encounter this sort of internal messaging feature on discussion forums. When a member has a new private message to read, the count of unread messages is indicated next to the private messaging link and they also get a popup alert message one time.

Once you have the switch on, you can also choose which usergroups can use private messaging and rate-limit their messages to prevent spam by using the Manage Usergroups page at Admin Panel -> Members -> Manage Usergroups.

In the member profile there's an option to select to accept private messages from everyone, no one, or just from buddies if the buddy list switch is enabled. WIth this, each member can choose how easily to be contacted. You can set the default value in the registration defaults, at Admin Panel -> Members -> Registration Defaults.