To archive messages first you need to create a new folder (or folders) to store your archived messages. Hopefully the screen shots will help to illustrate the process of creating folders and moving your messages to those folders. It's really not that hard once you do it yourself. Let me know if you have any questions.
1. On the left side of the screen in the option list under the "Your Control Panel" heading in the section marked "Private Messages" select (left mouse click) "Edit Folders". (Note: do not click "Edit Folders" in the section marked "Subscribed Threads")
2. A new screen will open like the one pictured below where you can create your new folder(s) to use for your archived messages. In one (or two) of the empty boxes type the name(s) you'd like to use for your new folder(s). I created two new folders. One called "Inbox Archive" to store messages I've received and one called "Sent Items Archive" to store messages I've sent.
3. After typing in the folder names you'd like to use, click on "Save Folders" at the bottom of the screen. You should now be taken back to your Private Message list.
4. Use the check boxes on the right side of each message to select the messages you'd like to move to one of your new archive folders. You can choose one message or multiple messages.
5. a) At the bottom of the screen, use the drop down menu for the "Selected Messages" options and select "Move to folder...". I believe this is the default setting so no action may be needed here for this process. This is similar to the process used for deleting messages as Sophia described above. If you'd like to delete messages rather than archive you'd select "Delete".
5. b) Click the "Go" button to perform the desired action. The "Move Private Messages" screen should appear.
6. Use the drop down menu to select the folder where you'd like to move the selected messages to.
7. Click the "Move Items" button at the bottom of the screen.
8. In the future, when you want to view the messages that you've archived, you can navigate to your new folders by selecting any of the drop down arrows that I've marked with an "8" in the pic below.