An International Standard Book Number (ISBN) is a code used to uniquely identify a book together. It also uniquely encodes the book's publisher and includes information about its language of authorship. The original 10"digit" ISBN10 (where a "digit" consists of a decimal digit 09 for the first 9 places and 09 or X for the tenth place, corresponding to a mixed base string), in use for more than 30 years, was officially replaced with a 13digit ISBN13 (where each place is truly a decimal digit) as of Jan. 1, 2007.
The digits of an ISBN are arranged in four groups (for an ISBN10) or five groups (for an ISBN13), which are sometimes (but not always) separated by hyphens. At present, an ISBN13 is always prefixed by the digits 978 (US ISBN Agency). The first group in ISBN10 or the second group for an ISBN13 is a single digit which encodes country or language in which a publisher is incorporated: 0 for English, 2 for French, 3 for German, 4 for Japanese, 8 for Indian publishers, etc. The next group of digits specifies the publisher, and may range in length from two to seven digits, with fewer digits used for larger publishers. Some publishers with offices in more than one country (at least when different languages are spoken in those countries) have multiple publisher codes and initial digits.
publisher  publisher block 
AddisonWesley  0201 
American Mathematical Society  0821 
Birkhäuser Basel  37643 
Birkhäuser Boston  08176 
Cambridge University Press  0521 
CRC Press  08493 
Dover  0486 
McGrawHill  0070 
Oxford University Press  0198 
Springer Berlin  3540 
Springer New York  0387 
Tarquin Publications  0906212 
Wiley  0471 
The next group of digits specifies an individual book, and may be from one to six digits in length. The actual number is eight minus the number of digits in the publisher group, so that small publishers may have only 10 books while large ones can have up to a millions books. The last digit is a check digit which may be in the range 09 or X (where X is the Roman numeral for 10) for an ISBN10, or 09 for an ISBN13.
For ISBN10, the check digit is computed from the equation
(1)

For example, the ISBN10 for the first edition of the printed version of MathWorld is 0849396409, and
ISBN10 
(2)
 
(3)
 
(4)
 
(5)

where denotes a dot product and is the vector composed of the first 9 digits of the ISBN10.
The scheme used by 978 and (future) 979prefixed ISBN13, is instead given by
(6)

(Book Industry Study Group). Therefore, the ISBN13 corresponding to the ISBN10 above would have check digit
ISBN13 
(7)
 
(8)
 
(9)
 
(10)

and so would be 9780849396403.
The ISBN is errordetecting, but not errorcorrecting (unless it is known that only a single digit is erroneous). The ISBN detects any singledigit error, as well as most twodigit error resulting from transposing two digits.