This button shows the status of the ticket and could be used to update status. Clicking the button opens a popup displaying a list of available statuses. Select Status in dropdown and click Ok.
Note that this action don´t save the ticket, it only updates the button and the Save button in lower right corner to the status you have selected. To actually save the ticket with the new status you have to use the Save button, or shortcut CTRL+ALT+U or CTRL+ALT+S. In this case both alternative shortcuts will execute the same action.