The most basic piece of software you need for editing--and the piece which editors are most likely to have--is a word processor like Microsoft Word. Word processor users inevitably end up saving multiple versions of the file at different times during the editing process; this allows them to quickly return to an earlier version.
Many books require illustrations or photographs. New material is created during the original writing process; then during the editing process, extra material is created and all of it is put into final form. Two types of programs are used for these processes: Vector art programs (like Adobe Illustrator) are used for line drawings, and digital editing or "paint" programs (like Adobe Photoshop) are used to prepare photographs and bitmapped images.
While some self-publishers use a word processor as their primary design program, such a book will not look like a professional production. A book design program like Adobe InDesign allows you to see how each individual page will look and allows you to control and alter a page in minute detail, right down to the spacing between two individual letters in a single word. While such control may sound like overkill, a few such editing changes can make a self-published book look like it came from a major publisher.
Most printers will prefer to work with an electronic production file of your book, and these files are almost always PDF files. A PDF file also allows an editor to see exactly how each page of a book will appear when the book is published, so a PDF file is an essential last editing step. There are many PDF creation programs available, but be sure to check with your printer before getting one; some publishers will not guarantee good results unless you specifically use Adobe Acrobat.
While the editing process can be linear, moving from word processor and art programs to book design/layout to PDF creation, the process is rarely so simple. Problems faced during the initial editing may require a writer to make new illustrations; choices made during layout may require a return to the word processor or art program in order to make them work; or a PDF file may expose an unexpected error in layout. This give-and-take is what makes book editing software so powerful.