Peer reviewed refers to a process through which most scholarly academic articles are vetted prior to publication. Although many books go through some type of review process, most books are not clearly identified as peer-reviewed or non-peer-reviewed; determining what type of review process a particular book went through is more than most professors intend for you to do. Your professor probably wants you to have 20 references using a combination of books and peer-reviewed articles.
When choosing books to cite, look for books that include footnotes or bibliographies citing the sources for their information. Books without sources cited might be considered to be equivalent to general magazine articles. If a book contains entries by multiple authors there may be a list of expert reviewers who were responsible for assessing particular sections of the entries. You might also look for reviews of the book in scholarly journals, which might provide a descriptive evaluation regarding the quality of scholarship in the book.