RICHARD BUCKMINSTER FULLER said: "... But when I have finished, if the solution isn’t beautiful, I know it is wrong." Yes, good design should be elegant, simple, beautiful and self-evident. You know it when you see it. Good designers know what to include and what to leave out of a solution. Less is more - unless its necessary!
Some of the factors which go to making a good website design are Page Layout, Browser and Device Compatibility, Site Navigation, Color and Graphics, Content Presentation, Functionality, Accessibility and Maintainability. Some of these areas will require more of your attention than others. But they all need to work together correctly. Just as one weak link in a chain can render the whole ineffective, so an ill-conceived part of your website design can undermine all the other efforts to make it a success. A professional can help you ensure they work together seamlessly.
"People ignore design that ignores people" - Frank Chimero The truth is people lack time and patience. With so many things competing for our attention there is little patience for designs which don't make it easy for the user to 1. know what the site is about, and 2. how to buy, download, browse or whatever. Make it as simple and self-evident as you can. See your site through the eyes of a first-time user. Is it clear what you are doing or representing? Your design should knock 'em out!