There are a few methods you can use to determine the edition of a book (but, remember that it is wise to double check by confirming true dates of first editions by consulting an author's bibliography, many of which are available online and can be found by using a search tool/engine - The usual methods of identification are as follows.....
a) Firstly, and most common - publishers usually use a numeric system for identifying the Edition of the book.First Edition Books
You will usually find a string of numbers (1 2 3 4 5 6 7 8 9) which indicate the edition number. For instance, if the numbers from 1 to 9 are shown, you have a First Edition (1 represents the First, 2 represents the 2nd, 3 represents the 3rd and so on, some books will show edition numbers nearing 100, but not starting from 1). Whereas a second edition would show 2,3,4,5,6,7,8,9 (the number 1 has gone). Some publishers show the numbers in a different form (such as 987654321 or even 135798642) or use letters (such as abcdefghi). There are other rare exceptions to this rule (such as Random House who instead state that it is a First Edition and start the number line at 23456789). However, some publishers do not use the number/letter line method at all, and there are some publishers who use the number line to indicate a first by their own publishing company, and not a true first of the title being described (you should consult an author's bibliography to confirm this, many of which are freely available on the internet and can be found by using a search tool/engine).
b) - many publishers simply state that it is a 'First Edition', or 'First Impression', or 'First Printing', and do not include the edition numbers as above.
c) - some publishers simply state 'First Published' followed by the year (e.g. 'First Published 1983'), and if there are no further printings indicated with subsequent dates, then you may well have the first edition .
d) - a few publishers make no distinction at all and further information or verification would need to be found by searching through an author's bibliography (freely available around the internet or in your local library) which will state the true first edition date and publisher of the book. Let's face it, it is always worth checking, and the internet is awash with free bibliographic listings, so take that extra few minutes to check, and check again (you wouldn't want to be mistaking a Book Club Edition for a true First Edition, would you?)
e) - some publishers quite simply state that there were no further printings, in some cases the letters 'NAP' will be present ('NAP' is an abbreviation of 'No Additonal Printings').
f) - there are some rare exceptions to all the above rules, where there is absolutely no distinction made whatsoever by the publisher. In this case, further investigation of the author's bibliographies is an absolute must together with any points of issue (particular points in or about the book that distinguish its edition - typographical errors would be one example). A bibliography lists an author's books with the dates and publishers details (and often some personal history of the author).
Note: there are some unscrupulous book sellers that state a book as a First Edition, when in actual fact it really is not (for example Book Club Editions/Pulp Editions which are budget reprints even though they may say 'first printed'). Any reputable book seller will always include at least the Publishers name and the date of publication, as shown on the Copyright Page. You can verify this with corresponding information in bibliographies, plenty of which is freely available around the internet or in your local library.