An ISBN is an International Standard Book Number. Every book sold anywhere in the world should have a unique ISBN, meaning that you can find any book in the world simply with that one number, which never duplicates.
Starting Jan. 1, 2007, all ISBNs are now assigned using 13 digits instead of the former 10 digits. This "new" ISBN format actually dates back many years, too. Books have long carried two barcodes: one of the 10-digit ISBN (now called ISBN-10) and another for a 13-digit conversion of the ISBN (now called ISBN-13) into an EAN (European Article Numbering) code, a worldwide standard—despite its name—that's used for retail stock-keeping and sales.
With ISBN-10s, only the first nine digits are meaningful; the tenth digit is a special code that's a mathematical calculation of the first nine digits known a a checksum. An ISBN-13 until now always started with 978, the prefix for books, had the same next nine digits, and then an EAN-specific 13th checksum digit.
Nine digits aren't enough to count every book published in the last 30-plus years worldwide, and thus the switch to ISBN-13, in which a new prefix, 979, will be used to open up a whole new range of numbers. These 979 numbers can't be converted back to ISBN-10, and as a result ISBN-10 has become a thing of the past, supported by bookstores only for backwards compatibility.
We'll always accept ISBN-10s at ISBN.nu, as well as all new ISBN-13 codes.
Publishers request ISBNs from R.R. Bowker in the United States and from a variety of agencies worldwide. Publishers apply to these agencies to get a range of numbers with a unique prefix which varies in length. Because of mergers and acquisitions, some publishers have several, or even dozens of prefixes.
Publishers choose what numbers to assign to which books following their prefixes. ISBN have no relationship with subject matter, nor do the dashes mean anything. But because ranges of ISBNs are assigned, even though most ISBN-10s aren't used, they're still reserved and can't be used by other publishers. (The same thing has happened in Internet Protocol or IP addressing, by the way; the Internet is slowly moving from 32-bit to 128-bit IP addresses, or from IP version 4 to IP version 6.)
All 978 ISBN-13 codes can be converted back and forth with ISBN-10 codes; when a 978 codes is used, books will carry both ISBN-10 and ISBN-13 information.
Every ISBN should be unique. Occasionally, ISBNs are reused for books that have long been out of print, but this practice is generally discouraged.