Can you move books to the different readers or do you read from you bought it?
Depends on the reader and the book.
Many of the readers use the standard ePub and PDF formats. If the book has no DRM (copy protection), then you can move it. So, I can move some of my books between say, the Nook, iBooks, or Stanza.
Kindle is currently odd duck out. It has it's own proprietary format, and does not support ePub. It does support PDF, but only on the actual device. So even if you get a Kindle book without DRM, possible, you can't read it in other book readers.
The exception is Stanza, which will/should (not tested) read Amazon's azw format if you rename it to .mobi or .prc. But only the unprotected books. The opposite is also true. You can rename .mobi or .prc files to .azw and read them in the Kindle app (tested with some minor problems several months back). You may not even have to rename them to work on the Kindle device itself. Haven't tested that in over a year.
You get the feeling that Amazon wants to leave a few reasons for buying their Kindle reader?
Most paid for books in any of the stores have DRM. Free books, like the Gutenberg Project books almost always do not. And you can find DRM free books on many other sites, though it takes a bit of looking.
So, not easy yes or no answer. Instead, you get a qualified, sometimes.
Disclaimer: Some details may be out of date, but the overall picture remains true. Things change constantly.