Kindle will only read Kindle books - sorry.
For e-books with DRM (Digital Rights Management, or copy protection) — which covers all of the major books — Amazon forces you to buy them from Amazon.
There are e-books without DRM available, and those you can buy from just about anybody. If you can't get them in the MOBI format that Kindle uses, you can convert them using a program like Calibre. Some sources of DRM-free e-books are Smashwords, Baen, Carina Press, Samhain, some of the tech publishers (O'Reilly and Microsoft, for example), and many of the erotica publishers. And then there are the free public-domain e-books from places like Project Gutenberg. The Calibre site has just started putting together a catalogue of DRM-free e-books that you can get from various places.
I'm new to online books. I just received a Kindle. Can I download Nook books for the Kindle or do I specifically have to order Kindle books for it?
One of the major drawbacks with the Kindle is that it is a closed ecosystem. You cannot buy a book at another publisher, and put it on your Kindle. You cannot buy an eBook from Amazon and put it on your Nook.
Kindle books must be read on either Kindle hardware or a Kindle app.