Nowadays ebook retailers offer (typically) thousands, if not hundreds of thousands, of free books, usually as an incentive to attract customers who will, presumably, spend money on reading devices, 'paid' books and other merchandise. I'm going to assume that regular customers know how to find the free ebooks at their retailers' websites (e.g., by using each site's bestseller lists and search engines).
I want to make sure that you also have access to a few websites that offer free ebooks in multiple formats. (Note, however, that not all books are available in all formats.) Books are typically in the public domain (more about this later), and all are offered 'DRM-free', i.e., do not contain additional code to prevent file-sharing. (Among other things, this means that you can use software like Calibre to convert files to different formats.)
First, what I'll call 'the Gutenberg family': the Project Gutenberg sites for the US, Canada and Australia. Think of the US site as your free download site of first resort, and the two others as complementary sites that focus on filling in gaps left by the US site. (The US site offers only books that are in the public domain [i.e., no longer under copyright] in the US. The duration of copyright is different in different countries. In general, the laws of the country in which you are located determine what it is legal for you to download.)
Each Gutenberg site includes information on ebook formats (generally EPUB for reading devices other than Kindle), the mechanics of downloading, and the conditions under which it is legal for you to download ebooks from the site.
There is also at least one French-language site with an admirable breadth of offerings: Ebooks Libres et Gratuits. Most users will need to go to the search engine for the site, which can be found here. (Note that files labelled 'Mobipocket' can be read on a Kindle.) All materials are in French; they may or may not be in the public domain in your country.
Let me add one final link--a site where I frequently find things that are available for free nowhere else: Munsey's. Munsey's is where you'll find more free pulp fiction and more free Henry James--and quite a bit in between--than anywhere else on the Web. The database may not have the sleekest interface you've ever seen, but there's a lot to love at this site.
One tip, applicable to all these sites: if you're going to be reading your free ebooks on a tablet, it may be easiest for you to browse these sites and download your books using the same tablet you're going to read on. This saves you one subsequent file transfer (e.g., laptop to tablet).
Also, some ebooks, conspicuous classics, are just elusive. You may want to use your usual Web search engine... always including, of course, a term like 'EPUB', 'Kindle' or 'ebook' with the title.
Finally, you may want to investigate ebook lending (friend to friend, as offered for Kindle and nook users), and borrowing from public libraries (long available via nook and recently announced for Kindle users in the US as well).
As always, I encourage you to use the comments function below. Be sure to mention your own favorite sites....