Simplicity
Unless a website exists purely for entertainment, visitors want to get to the information they need as quickly and easily as possible. They are irritated by unnecessary frills that get in the way of the content. Simplicity is the watch-word for accessible and usable websites, and it's what users want. Simple websites focussed on content are also the most visible to search engines.
Cross platform
While the majority of website visitors are using a desktop computer or laptop, increasingly mobile devices are being used to view web content. As an iPhone user, I am always pleasantly surprised when a website has been optimised for small screens and slow data connections. But it's amazing how many are not – a glaring example is Vodafone, which of all websites one would expect to be mobile friendly!
You'll notice that my own website is mobile friendly, of course!
I've taken care to make my own website mobile friendly. Try it on a smartphone – scan the QR code below if you have a reader.

Robust
Back in 2004 I designed and built a website for Lambs Bricks, which is still in use today. The fact that the extensive javascript powering the site still works with zero modification is a testament to the robustness of my coding back then. Things have moved on considerably now, and there are various Open Source javascript libraries (e.g. jQuery and prototype.js) out there to make cross-platform coding not only easier and more robust, but also more dynamic and powerful.
Future proof
HTML5 is coming, and while not fully supported by browsers at present, within a few years it will become a usable standard. While one cannot write pure HTML5 and expect it to render properly everywhere at present, it is wise to start using it's features wherever possible.
Websites based on a content management system such as Wordpress or DRUPAL are becoming the norm, as website owners want to have the ability to update their sites without being tied in to a website design company.