When not otherwise specified by the user, a search term with multiple words will by default AND those words together. Any item in the results must include "Christopher" AND "Hill." The boolean operator AND will always be a narrowing influence because ALL words in the term must appear in the document. Of course, when you are searching full-text databases and/or documents with millions of items, a simple AND search might not seem to be limiting your results, but if you were to search for "Christopher" without adding the word "Hill", or vice-versa, your search results would almost certainly be greater than when using both terms.
OR, on the other hand, will increase the "comprehensiveness" of the results over using any single term. The document either contains the word "Christopher" OR it contains the word "Hill" (or both). If you find that your search is too narrow, you can expand it by using OR. For example, a search for:
Author="Christopher Hill" AND Title=("Law" OR "Ships")
would pull up Christopher's book on Maritime Law and his book on Arrest of Ships. Note the parenthesis around the OR phrase. When the advanced search of a database allows you to specify different fields in which to place your terms, you would likely not need parentheses. If, however, you were searching in a single box, your results could be very different without parentheses.
Author="Christopher Hill" AND Title="Law" or "Ships"
Would retrieve anything by Christopher Hill with the word Law in the title, but it might also retrieve anything with the word Ships in the title, whether or not it was authored by Christopher Hill. The parantheses ensures the logical groupings.
Not might not alter your search results, but when it does it will limit them. If for example, you searched:
Author=Christopher Hill NOT Publisher=World Maritime University
The results would be the same, because there are no works authored by Christopher Hill and Published by WMU. On the other hand:
Subject = Piracy NOT Copyright
Would likely limit your search results in an appreciable and relevant way.