If I have a kindle paperwhite, am I limited to books from Amazon? Can I purchase or download books from other sources/retailers?
Kindle devices allow you to load non-Amazon files onto them; you can copy them directly to your device using a micro-USB cable, or have them sent to your device wirelessly, through Amazon's Kindle Personal Documents service.
You are limited to the types of files supported on the Kindle: at present, these are
.azw, .azw1, .txt, .mobi, and .prc. In addition, Kindle can read
It is extremely easy to convert files into Kindle-supported formats from other popular formats -- such as
.epub files, Word documents, and web pages. Amazon's Personal Documents Service can help you with transfer and conversion, and Calibre is an extremely popular tool for ebook-management which provides such conversions easily. PDF files which are mostly text are also quite easy to convert to a supported ebook format.
The one big wrinkle in the all-around compatibility I'm describing here is that books protected by DRM (a Digital Rights Management scheme), the DRM prevents it from being converted into different formats. So an ebook you buy at barnesandnoble.com, which is formatted and protected exclusively for the B&N Nook e-reader, cannot be easily converted for the Kindle.
Many major ebook retailers rely on DRM to help "lock you in" to their specific device, platform, and store. So basically, ebooks you get at non-Amazon stores are unlikely to be compatible with the Kindle (unless they specifically note that the ebook has no DRM), whereas ebooks from other sources (e.g. the out-of-copyright classics at Project Gutenberg, or ebooks you create yourself using a service like Instapaper) should present no difficulty.
As far as I know (but I'm not a kindle user, so I can't tell you first-hand), you can side-load books from other sources (i.e. Project Gutenberg or MobileRead), privided that they are in a format recognized by the Kindle (they should be MOBI and AZW3); if you have books in other formats, you can easily convert them by using Calibre.
In my experience I'm able to download .mobi files directly to various Kindle models. The web site needs to include appropriate meta-data with the HTTP response (which includes the ebook).
As an example try the following web site http://www.munseys.com/ and one book I picked from the homepage http://www.munseys.com/book/35352/Red-Headed_Sinners that includes the KindleFire format (actually a .mobi file)
I don't currently have a Paperwhite with me (I can try later on for you) to test whether this book downloads correctly.
Also, since other people are commenting on the Kindle Fire being more restrictive, here's an article that includes 3 ways to get books onto the Fire http://google.about.com/od/kindlefire/a/How-To-Put-Non-Amazon-Books-On-Your-Kindle-Fire.htm
My experience - I use a Kobo, not a Kindle - is that ebooks available from outside the respective stores are usually available in multiple formats to accommodate different devices - e.g. Mobi for Kindle, ePub for Kobo, PDF for computer. Various publishers/authors provide ebooks in more than one format - see O'Reilly, SitePoint, LeanPub, etc. for examples.
All of that to say, you're not limited to Amazon (or an immediate store if using a different platform) for sourcing content.
There is a utility from Amazon called "Send to Kindle" that allows you to send your mobi files converted with Caliber directly to your kindle over WIFI.