Most for-purchase books are protected by DRM (Digital Rights Management) systems that are particular to the ereader, so, no, you can't count on being able to buy Borders ebooks for your Kindle, or Amazon ebooks for your Kobo. There are lots of places, however, which offer free, legal ebooks that will work on most ereaders. Sometimes it's a single format that most support (EPUB, TXT, PDF, etc); sometimes multiple ebook formats are listed per book.
The 13 sites listed here (Patricia Clark Memorial Library, ManyBooks, Project Gutenberg, Smashwords, Feedbooks, Baen, Amazon, Barnes & Noble, Google ebookstore, Internet Archive, Open Library, Sony Reader Store, and Kobo eBooks) have legal, free ebook downloads:
http://www.howtodecide.com/ebook-download/
Also, if you master a good conversion tool such as Calibre,
http://calibre-ebook.com/
you'll gain access to more formats than just your ereader's native ebook file formats. Calibre will not, however, overcome DRM of protected for-purchase books.