Check the author's bibliography for original print run dates. You can often find bibliography information at an author's online website, or in your public library. Books are usually listed by first print run date, so if you come across a copy dated after the original print run, you know it is not a first edition.
Review the copyright page for a numeric line that states the print edition. For example, you may see a string of numbers like "1 2 3 4 5 6 7 8 9," and the one being present would indicate a first edition if the copyright date at the top of the page corresponds with the author's bibliographical information.
Check the copyright page for the words "First Edition." Some publishers simply add the words "First Edition" to the copyright page. You will want to confirm the publication date with the authors bibliography, as sometimes another publisher will republish books after the original contract runs out.
Look for a "First Published" date. Some publishers will list a string of publication dates, but if there is only one date in the margin, it is likely the edition you hold is a first edition. For example, you may see: "First Published 1976." If no other dates follow, you may have a first edition. Double-check the author bibliography to confirm the original print run date.