Holy cow, I wrote a book!
On the copyright page of a book
(typically the back of the title page),
you'll find a row of numbers.
Something like this:
Printed in the United States of America
10 9 8 7 6 5 4 3
As Dave Taylor explains,
the smallest number tells you which printing of the book you have.
For example, if you see "10 9 8 7 6 5 4" then you have a fourth printing.
Dave doesn't explain why printers use this convention, however.
I forget where I learned this; I think I read it in one of Don Knuth's books.
It has to do with how books are historically made.
Each page of a book is converted to a metal plate which is used
to make impressions.
If another printing run is necessary,
you load the plates back onto the printing machine and off you go.
But how do you indicate that this is a second printing?
It would be expensive to burn a brand new plate just to change
the word "first" to "second" on the copyright page.
Instead, you pre-load all the printing numbers onto your master,
and each time you start a new printing run, you scratch off
the lowest number.
Even though a lot of book printing nowadays is done with computers
rather than metal plates, the old method of indicating a printing is
retained out of tradition.