Search your local library system's catalog for an electronic edition of a particular title. If it's available, the library website should offer a link to the appropriate reader software. Otherwise check listings for the publisher or for dedicated e-reader devices.
Download the application software to your computer over your regular Internet connection. It will probably also set up a file to organize your e-books as you collect them.
Return to the bookstore or library site and follow instructions to add the e-book file to your cart and check out. The book should appear in your "downloads" folder, or you may be asked to direct it into the file for your new reader application.
Open the reader application and find the new book. Each reader has its own method for organizing your e-book library, remembering your place in each book and tracking your bookmarks and annotations. E-books you borrow from the library in Adobe EPUB format may simply expire when your borrowing period is up, and if you attempt to open the file still showing in someplace like your "downloads" folder, you'll get a message saying your system is unable to find the license. Delete that file, and if you haven't finished the book, return to your library site and borrow it again.