To archive a conversation, go to your main messages page and click the "x" next to the message. The conversation’s history will be preserved, and you will still be able to find it later. If the same person sends you a new message later, the archived conversation will reappear, and the new message will be added to it.
To permanently delete a conversation from your messages, simply open the conversation, then select "Delete Messages..." from the Actions drop-down menu.
If you click "Delete All" at the bottom of the page, the full conversation history will be permanently cleared from your messages. You can also check the boxes next to individual messages and click "Delete Selected" to permanently delete parts of the conversation.