No author has to have an author website, but it's recommended for publicity. Twitter, Facebook, and Goodreads is also recommended. Social media is a good way to promote your book. You can use Google+ too, any social website is good. The publicity will be good for your book, as long as you mention that you've written a book (and put the title of it) in the BIO section of the website. You could even add a picture of the book cover to catch eyes.
Anyway, back to the author website topic. Having an author website for yourself is good for publicity, but also good for your readers. The people who have read your book, and liked it a lot, will want to know when your next novel will be released. Readers are usually very loyal. If you wrote a book that they liked they will check in on your website to see when your next book will be released and what it is about.
Take Sarah Dessen as an example:
(Sarah Dessen's books)
Sarah Dessen is the author of multiple teenage girl novels. Every time a new book comes out, her readers are waiting.
Self-published authors usually use blogs for their 'author website.' You can get ideas for your author website from other self-published authors such as:
Amanda Hocking -- author of the Trylle Trilogy: Amanda Hocking's Blog
Michelle Hodkin -- author of The Unbecoming of Mara Dyer: The Michelle Show
Tammara Webber -- author of Easy: A Room of My Own
Here are some blog websites that you could consider using for your author website:
Here is a comparison chart of some of the blogs. Pick the best one for you!
And here are some other links (yes, more links) that will help you get your 'author website' up and running!
13 Steps to Creating an Author Website Readers Will Love
Build a More Effective Author Website
10 Ways to Build Traffic to Your Author Website
Good luck building your website!!!