Choose whom you can receive private messages from
You can choose whether you are able to receive private messages. When you visit your settings page (click on the icon/avatar in the upper right and choose settings), you'll see options to allow private messages from no one, friends only, or everyone.
Send a private message
To send a private message, hover over the other member's name. If that member has chosen to allow private messages from everyone, an option to send a private message will appear in the pop-up window. You can also click on the connect button on the other member's profile page, and select send a private message from there.
Either one of these actions will land you at your messages in the site banner. (You can also reach your private messages directly by clicking the message icon in the banner.) There you can send a message, or view other messages that you've received.
Delete private messages
You can clean up your list of messages by selecting any private message you'd like to remove and clicking hide. The message will reappear if the other member sends a new message. If all participants hide the message exchange, it will be completely deleted.