The Eye is a counter showing views number of your entry including entry page, recent journal entries and friends feed when lj-cut has been opened (including unauthorized visitors). Every view of visitor will be counted, e.g. if user will open lj-cut, go to comment thread, add comment and refresh page it will generate 4 views.
Where I can see this counter?
Counter is visible for logged users only on entry pages in system style, in posts created in the new editor
, on feed pages and in embedded entries.
You can enable or disable the counter at settings page