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The New Mailbox Search dialog has a series of options. The administrator can legitimately recover mailboxes for users' who have lost their data, or they can search all the mailboxes across the organisation. This could have an important function if there is an epidemic of swearing at a K12 institution, for example, or bullying and suchlike. In a university setting where academic freedom is a right as well as a privilege, perhaps not.
The Search name is the name of the folder in a separate 'discovery' account where the mailbox search is stored, and in Exchange, the separate account where the results are stored is in the standard Discovery Search Mailbox.
Colour key for viewers: Red border is the administrator's email view. Green border is the student user's email view. Blue border is the administrator's console.
For more background on this screenshot gallery, head over to the iGeneration blog which explains all.