The first place you need to look is the copyright page, just after the title page and before the contents of the book. In there, you see a lot of useful information such as the year it was originally copyrighted, who the copyright owner is (typically the publisher or author), the ISBN number, and the Library of Congress cataloged information. Unless the book has been privately printed, it has some kind of identification on one of the first pages.
If the book has an ISBN (an identification number) or the equivalent in your area, you can probably look it up in Books in Print or a similar publication.
Practically, you can confirm this by contacting the publisher listed in the book, if possible.
Library of Congress and the US Copyright Office work in conjunction to create a searchable database for books. For books published after 1975, you can visit http://cocatalog.loc.gov. You enter an author or title and see the registration number and the year that the copyright was registered. For older works, you can contact the Library of Congress directly and either pay them to do a search for you or do the search for yourself.
There are also resources that list books in the public domain.
You can also ask a reference librarian at your school or a local library.
If you need help researching the copyright history of a book or any work, feel free to contact New Media Rights at (619) 591-8870 or email@example.com for free, pro bono legal assistance.