This is not as simple as it should be. I do cover modern firsts in more detail in my How to Spot a First Edition Harry Potter book.
Many modern books use a number line on the copyright page, with the lowest number (e.g. 1) indicating to which print batch a book belongs.
Unfortunately, it's not always that easy.
Different publishers use a variety of methods, none of which are particularly intuitive. Your best bet is to do your homework before you buy.
No Number Line? No Problem!
For most fiction published within the last twenty-five years, determining whether or not a copy is a first edition, first printing is a doddle, just check out the number line.
But, what about older books and publishers who refuse to use a number line?
What if there is no printing indication at all?
What if it states "First Edition" on the copyright page? Does that means it was also a part of the first print run?
Without a doubt, determining edition can be very tricky!
Colophon (logo or publication history)
Colophon has two meanings, both relevant to this discussion. Colophon can refer to the publisher's emblem or logo. Some publishers print their colophon (logo) on the copyright page or at the back of the book to indicate a first edition. Colophon may also be defined as an inscription detailing a volume's publication history that is usually found at the back of the book.
The same date appears on the copyright page and the title page. Scribner's used this method in some pre-1930 volumes.
No Additional Printings
With this method, if there is no mention of subsequent printings, the book may be assumed to be a first edition, first printing. This method is widely used.
First Published, Year
Some publishers state "First Published" followed by the year or the month and year of publication. If no additional printings are mentioned, you may assume that the copy in question is a first edition.
A publisher may state "First Edition," "First Printing," or "First Impression" on the back of the title page, or on the copyright page.
No Designation for First Editions
A publisher may choose to only identify later printings and editions. Thus, a first edition, first printing copy would have no designation at all.
I do hope this helps you identify your book. You can also go to the library and borrow a bibliography of the author in question, or pick the brain of a kind-hearted and knowledgeable bookseller !