This article was written using a Kobo (Clara HD) to present the use of a web browser with an eReader. Therefore the screenshots presented to show how to use the browser are related to the Kobo interface. Most eReaders offer a browser but its location may differ from one eReader to another, so please refer to specialized articles on your eReader if the present guide is not enough to find it. If your eReader don't offer any browser, you can't use this guide.
With your eReader you can download an eBook directly from Internet. To do this, you need to use your eReader's web browser to find a website that will offer you a direct download of an .epub file. The goal is to perform almost the same action as when you download a recently purchased eBook from your eReader's internal store, if your eReader offers one.
From the Kobo browser example, from the eReader's home screen, tap on "More" in bottom navigation and then choose "Beta Features"...
...and tap on "Start" in "Web Browser".
This will open the home page, usually on Google, you have URL bar at the top, back, next & refresh buttons at the bottom with a zoom bar, don't leave out the three dots indicating additional options at the bottom right as you can add websites as favorites.
You can now go to any website with your eReader's browser. You have to find websites allow direct download of books and you can direct download eBooks.