You've all seen them, they're the 10-digit numbers on the covers or backs of every book published in recent memory. Sometimes, they are incorporated into UPC or EAN bar codes (used for product identification at check-out).
The basic format is 10 digits (the last one can be 0-9 or X). There are four parts to an ISBN, and the standard mandates that they be separated either by a space or a hyphen. The numbers are assigned in a heirarchy and each jurisdiction can decide what groupings the numbers should be separated into.
To be certain that the numbers are valid when read, the last digit is a check digit. To compute the checksum, each digit in space N (starting at the left as 0) is multiplied by 10- and added together. The final digit, if 'X', is added as 10. Next, the sum is divided by 11 and any remainder indicates that the number is invalid.
For those curious about the EAN translation, EANs are created by placing 978 in front, removing the last digit of the ISBN, and then computing a new check digit and appending that.
On the UPC front, there is no representation of ISBNs in the UPC standard; however, the UCC (Uniform Code Council, the folks who rule the UPC and EAN) have set a date of January 1, 2005 for retailers in the US to begin using the EAN instead of the UPC (distinguished by the 13-digit instead of 12-digit code) and at that time 10-digit ISBN will be replaced with 13-digit ISBN, based on the EAN.