Most people know that there is a free Kindle app for iOS that will let you read books from Amazon's Kindle store. Apple also offers a free iBooks app. Fewer people have heard of Project Gutenberg, where you can download over 30,000 free books, and fewer still have heard of the Magic Catalog from FreeKindleBooks.org.
The "Magic Catalog" isn't where Harry Potter shops -- it's a "book" containing nothing except links to other books. You can use the search feature of your favorite e-book reader to look for books; with a few taps, you can have the new book on your iOS device without ever needing to go through iTunes and without spending a penny.
The easy way
If you already have the iBooks or Kindle apps installed, simply go to Magic Catalog on your iOS device and tap the appropriate link for whichever app you are using (MOBI Edition for Kindle, EPUB Edition for iBooks).
The slightly harder way
If, for some reason, you'd prefer to download the books to your computer and transfer them to your iOS device (or if you already have .epub, .mobi, or other e-books on your computer), you can do so using iTunes. If you have the Kindle app installed, it will appear under the "File Sharing" section in the "apps" tab, and you can add books there.
You may notice that iBooks does not appear in the "File Sharing" section. To transfer books to iBooks, simply drag to the "Library" section of your iTunes library (see this page at Apple.com for more details).
Two minor flaws
Kindle: When using the Magic Catalog with the Kindle app, you'll find one glitch in the process. When you tap on a book in the catalog, the Kindle app opens a blank page with an arrow at the top left. You have to tap on that arrow to open the page in Safari, which will then show you a link to open the book in the Kindle app. Then when you go back to the Kindle app, you have to tap the "Done" button on the top right in order to see the new book. It's a minor annoyance, at worst.
Kindle and iBooks: When you are on the web page to download the book, you will not see the name of the book; you'll see something like "pg23.mobi" followed by the size and a button to open the book in the Kindle app (or iBooks if you have the EPUB version). Once you open the book in the appropriate application, it will show the correct name.