Books from iBooks, Sony, or Nook ebookstores generally have proprietary DRM (digital rights management software) that allows the book to be read only on a specific device or app that is linked to that bookstore. Books from those vendors are also in ePub format (a standard format, but not supported by Kindles), as are many of the free books on the web. A great source for free, non-DRM, out-of-copyright books is Project Gutenburg, which offers ebooks in different formats. If an epub book doesn't have DRM, you can easily convert it to the Kindle format using Calibre, free ebook conversion and management software (they do rely on donations, so consider making one if you use it and like it). Calibre can convert ebooks to and from Mobi (Kindle), epub, RTF, PDF and other formats.
There are a few ebookstores that sell ebooks in Kindle format but without DRM, and these books are perfectly compatible with any Kindle. Two excellent examples are Smashwords and O'Reilly. Smashwords is an enormous ebookstore that sells nonfiction and fiction ebooks in pretty much every format, even PDF and plain text. O'Reilly sells technical books, and they also offer multiple formats, and a daily ebook deal. These vendors generally offer books in Kindle format, so you don't have to convert them. With Smashwords, you will need to download the file to your computer and then move it to your Kindle. O'Reilly offers an email option for Method 2, described below.
Moving files to a Kindle: Method 1
One way to do this it to download the book, and then connect the Kindle to the PC/Mac with the USB cable, and then drag the file into the Documents folder on the Kindle. The file extension should be either azw, mobi., or prc for the file to appear as a Kindle book when you open it on the Kindle. This method does not require wireless access and doesn't incur any charge from Amazon.
Moving files to a Kindle: Method 2The second method is simpler but does require a one-time set-up step; you can email documents to your Kindle. The first thing to do is to go to the Manage My Kindle page, accessible from your Amazon My Account page, and whitelist your email address, and any other email address that you want to be ale to send books to your Kindle. Every Kindle has its own email address, and to prevent it from getting spam, only email from addresses you have authorized will be accepted by your Kindle. So, first find your Kindle address by clicking Manage Your Devices on your Kindle page, left column, and make a note of it.
Next, whitelist the email address(es) you will use to send documents to your Kindle by clicking Personal Document Settings and using the link for add a new approved email address. If you buy books from O'Reilly or any other vendor that offers non-DRMed Kindle format ebooks and email delivery, you can add their email addresses and send books you buy from those vendors directly from their website without downloading first.
Once you have done that set-up step, you can send any file with a mobi or prc extension to your Kindle by creating an email, addressing it to your Kindle, and then attaching the file. You can also send personal documents (plain text or MS Word files) and PDFs using the same email method. If you have a PDF you want Amazon to convert to a Kindle book (don't expect it to be perfect), you must put the word ‘convert’ (and only that word) in the subject line of the message; if you don't, Amazon will load the book to your Kindle as a PDF and you won't be able to change the font size.
Once caveat to the email delivery is that if you use a 3G connection to deliver the book, Amazon will charge you 15¢ per MB for each file. There is no charge if you use wifi to send the books or other documents, regardless of file size.
One reason I want to be sure Kindle owners know about this feature is that it is much easier to make a book free in Smashwords than on Kindle. Even when Amazon price-matches (which they usually do only if they notice the book is free on Barnes & Noble), they only make the book free in the US Kindle store (as they did with The Sixth Discipline). Folks who are not in the US who want to take advantage of a book promotion can get it in Kindle format from Smashwords and send it to their Kindles. Just be aware that Amazon's tools, like the home screen display and the Manage Your Kindle settings, will treat any book you didn't buy from Amazon as a ‘personal document’ rather than as a book. If you filter your home page to show "Books only," ebooks you got from non-Amazon sources will not show up.
Note: Here's a more recent post on the same topic.