What is the meaning of the check box and circle in this iTune U
The checkbox is a tool for downloaded items to specify if you want that specific item to sync to your device. There is a setting in your iTunes device list to sync only checked items in iTunes:


If you have that setting turned on, unchecking any song or video will tell iTunes not to sync that item to your device during the next sync.

The circle is a progress indicator for downloaded items: 


A filled circle indicates that the item is unplayed.
A half circle indicates that the item is partially played.
No circle indicates that the item has been played to completion.