There seems to be two types of ISBNS for every book: ISBN-10 and ISBN-13. I wonder what differences are between them, and in what cases to use which? Thanks!
The original ISBN (International Standard Book Number) specification of 1970 defined a 10-character identifier for books (ISBN-10). In 2007, a successor specification defined a 13-character identifier (ISBN-13). The latter was created because of a need for compatibility with other international trade identifiers (like UPC), and because numbers were running out and additional ones would be needed.
ISBN-13 is partially compatible with ISBN-10. If a book is assigned an ISBN-10 number, then there is an equivalent ISBN-13 defined for it, which is "978" followed by the first nine digits of the ISBN-10 code, followed by a single check digit. For such books, the publisher normally prints both codes in places like the title page verso for the convenience of users who may still be using ISBN-10, and haven't fully upgraded to ISBN-13 usage. At some point the ISBN-10 numbers will be exhausted and new format ISBN-13 codes will be used, which will start with "979"; those codes will not have any ISBN-10 equivalent.
More infomation is available at, for instance, Wikipedia's ISBN article.