By default, you will have two folders in your private message area. These are the Inbox and the Outbox folders.
The Inbox folder contains all messages you receive - both new and old - along with the name of the person who sent the message to you, and the date and time the message was sent.
The Outbox folder contains a copy of messages that you have sent, if you have specified that you wish to keep a copy of the message for future reference.
Each folder has a set of message controls which allow you to select messages and then delete them completely.
You will need to periodically delete old messages, as the administrator has probably set a limit to the number of private messages you can have in your folders. If you exceed the number of messages specified by the administrator, you will not be able to send or receive new messages until you delete some old messages.