The answer to this is that it depends. What do you want to build a Web site for? Who is your audience? Is your Web site for personal purposes or professional aspirations?
Today, a whole lot of tools exist to make building Web sites a breeze. HTML editing software like Dreamweaver or FrontPage literally takes away the need for doing your own HTML coding. Dreamweaver and FrontPage both have easy-to-use interfaces, so most anyone who understands the basic menus in Microsoft Word can figure out these programs relatively quickly. And they're both "visual" programs, meaning they build the HTML in the background so you don't have to worry about it. Or, if you want your own blog site, check out one of the online tools like Blogger or Movable Type, which allow you to use pre-existing templates so you'll never worry about the code (or won't have to worry about the code until you decide to change your site design).
However, most Web designers still argue that you should understand some HTML, even if you choose to create your Web sites with one of the fine HTML editors. Why? Because understanding the basics — which HTML tags are necessary to create a bona fide HTML file, which tags are the most common and why, and how tags work in general — only helps you. One of the common problems Web designers cite about HTML editors (including both Dreamweaver and FrontPage) is that they create pretty messy HTML files. And the more mucked up your code is, the more bogged down your Web site will be, the more memory and bandwidth it takes up, and the slower your pages will load on the Internet (if they load at all). Knowing enough code to clean up your pages — and to fix errors your HTML editing software might even be creating — is important.
Of course, if you aspire to a career in Web design, knowing HTML is crucial. Would you go to a doctor who knows how to use the echocardiogram equipment but can't read the results of your test? A hiring manager in Web design would say the same thing about a person who can't build a page using basic HTML.