This chain of numbers lets you know which printing of the book it is. For example, if you look in a book and you see this series of numbers:
10 9 8 7 6 5 4 3 2 1
that tells you it is the first printing. If you look in a book and see this:
10 9 8 7 6 5 4
that tells you it is the fourth printing.
It is very common for a publisher to print only a few thousand copies of the book in the first printing. If that first run sells out, then a second printing is ordered to fill demand. Between the different printings, the author often has the opportunity to make minor corrections to the book. By identifying different printings, the publisher and the reader knows exactly which version of the book it is.
This system for identifying printing numbers has been around for a very long time. The main feature of this system is that the printer can make the change to the printing number without having to add anything to the page or re-typeset the page. Simply by erasing one of the numbers (or whiting it out or whatever), the printing number is updated.
When the author wants to make more significant changes in a popular book, the book usually comes out in a second edition. A second edition is an entirely new book, with a new ISBN, a new cover and a new card in the card catalog.