The law differs country by country, but based on when a book was written and when its author died, the book's copyright eventually expires. At that point, the book enters the public domain.
Translation? At that point, the book can be published with no payments to the author. That means a hard copy book can be published more cheaply, and full text versions of the book are likely to be available online and for e-readers for free.
All the children's books below are now in the public domain in the U.S. and, in fact, in most countries in the world. That means you can easily find them online at sites like Project Gutenberg, or often download them for free to an e-reader like the Amazon Kindle.
I'll make it easy for you to get started.
Below you'll find a list of classic public domain children's books.
(I'm going to limit it to one from each author. Explore that author to
find more of their books.) I'll link you to an online version of each
book, as well as a free Kindle version (or darn close to free. You'll
have to click to be sure). If we have additional material on the site
about the title, I'll link you to that too. Enjoy these classics!
(Webmaster's note: Many of the free Kindle versions lack illustrations. Generally, by spending 99 cents, you can get an illustrated version.)
The National Endowment for the Humanities' classic children's literature books list.
Best Children's Books - Find, Read or Write home page.