The Search and Clear buttons activate your search and clear out the search components respectively.
The interactive search components (People and Organisations, Work, Keyword, Subject and Scope) may still have previous search terms in them even if the drop downs are closed. Use Clear before you start brand new searches to ensure a clean search.
Clicking the plus (+) button on search boxes adds another search box for immediate use in an AND search combination.
The downward arrow that appears when you add another search box lets you toggle between 'AND' and 'OR'.
Now you can see two kinds of minus (-) buttons.
Both boxes feature the (-) button on the left. Clicking this turns the box red to indicate the word contained will be excluded from the search (a NOT search).
The (-) on the right will remove the search box.