In this case the first edition refers to the original hardback.. It will either have 'first edition' printed on the back of the title page or the list of numbers printed will end with a 1. It would appear that your book is a first edition of the paper back version, which isn't quite the same thing.
You open the book to the page where copyright information is printed. That page is usually one of the first 3 pages in any book, and is printed on the left hand side. It will be full of information in small font, about the publisher, and author and copyright and such.
One of the pieces of information will be the edition. For example, I'm looking at a random book now (not HP). It says "First Simon & Schuster paperback edition 2005"
So first of all, if it's a paperback, it will not be a first edition for sure. Paperbacks are usually printed last. You need to know a year the very first HP was printed, and if you have a hardback book that says first edition + that year, then you've found it.
Good luck. :o)
First editions refer to the limited runs of initial hardback copies. I think there were only abount 500 hardback copies in the first run of the first HP book. They are worth a stack nowadays....