Find the book's copyright date and publisher. Look for this information opposite the title page. If the book was very popular, there may have been subsequent printings to meet demand. This would be evident by the publication data, which could also include the month of the printing. Publishing practices vary widely, and the only way to be sure that you have a first printing is if it's noted in the book, or if the edition was numbered and you were able to find out from the publisher or its distributor how many books were in the first printing.
Avoid book club editions. These are "secondary market" editions, published after a book became popular. First editions will have a price on the dust cover, but book club editions will not. Book club editions are often smaller and printed on cheaper-grade paper stock. They may have the name of the club, such as Book of the Month or Literary Guild, on the inside of the dust cover, too. Club editions also may have unique letters or branding above or below the publication date.
Examine the publication data. Some first editions will actually say "First Edition" or "First Printing." Most books published after 1968 (and all after the 1980s) have a universal publishing code shown as a row of numbers at the bottom. The smallest number is the printing. The copy of "Pioneers of the Air" shown here was published in 1991, is a first edition and a first printing. Older books may be more difficult to place. The book shown in the title and first section have matching publication and copyright date---1900---on both title page and reverse, indicating this is a first-first---a first printing of a first edition.
Look on the cover page, which is right before the title page. If there's an inscription by the author, your book is likely a first printing.
Enlist help. A collector's club or antiquarian bookseller can find information on a specific book, either on the Internet or in a number of references that will list how many books were printed, the composition of covers and binding, type of paper and any unique markings that would set them apart. Also to to the public library and check out a source like "Pocket Guide to the Identification of First Editions," 6th edition by Bill McBride (2000), or "First Printings of American Authors" (five volumes) edited by Bruccoli and Clark (1979).