ISBN-13 has 13 digits and starts with 979. ISBN-10 has 10 digits and starts with 978. Publishers realized that they would soon run out of the 10-digit ISBNs so a 13-digit ISBN was created.
In the 19th century the United States and the United Kingdom tried to develop a way of distinguishing books. After several different systems were used publishers settled on the ISBN-10. The ISBN was applied to books in the United States in 2007.
The last number of the ISBN is a checksum. This is a way of validating the ISBN. Each of the first nine digits of the ISBN, excluding the last number, is multiplied by a number ranging from 10 to 2, then added together. The sum of those numbers is then divided by 11. If that number doesn't have a remainder then it is a valid ISBN.
The first group of numbers in ISBN-10 or the second for ISBN-13 is a single digit. This tells where the publisher of the book is based: zero for England, two for France, three for Germany, four for Japan, etc. The next group of numbers identifies the publisher.
If you are a self-publisher you need an ISBN to sell on Amazon.com or at bookstores. There is a fee for registering one, plus a processing fee. You can apply online at the ISBN.org website (see Resources).