The copyright page of a book will give you the first clues to determine if it's the first edition of a book. Look for any sign, language, or marking that indicates a rare book is a first edition.
If it's listed, you can verify that the copyright date is the same as the book's first edition printing, which you can check online. You should also check the publisher and ensure it's the same as who first released the book.
A slightly more complicated means of determining if it's a first edition is to look at the number line or printer's key near the bottom of the copyright page. These lines can often contain information about the printing but can vary from publisher to publisher—if they're there at all.
A quick online search should tell you how to read the specific type of printer's key used in the book.