When I was doing HTML some years back the standard resoltuion that you would design for, "this site is best viewed in" was 1024 x 768. With the move on in times and computing power what is the standard resolution that websites are best viewed in and designed to these days ?

I wouldn't say there's a standard resolution. 1024x768 is currently the most used resolution, but you should still design your sites to look good across a variety of resolutions.

If your site is technical in nature chances are people will be using something larger than 1024x768, though maybe they're also viewing your site on their cell phone at 320x240.

If you're thinking about a fixed width site you're probably safe to target 1024x768, but there really isn't any single standard resolution.

Interesting point about the mobile phones, do people usually get round this by designing a specfic desktop site and then another site for mobile users that they can browse on their phones ?

Some do. I think if you build to the standards you're in decent shape right away. I try to keep mobile in mind so for example my own site I have the columns floating. If the screen with gets too low then the right hand columns drop below the content.

The site isn't specifically developed for mobile, but at least it still displays ok.

Recently I added a plugin (my site is on WordPress) that helps display the site on iPhones and I think some other mobile devices. I don't have those devices to test though so I'm not sure how it looks.

I think the key is to be flexible in your designs and realize there are many ways people might access it. And to test as many ways as possible.