Great question @Jo_G!
The deceptively simple answers are
1. Compelling imagery
2. Stylish copy
3. Perfect functionality
Great design articulates ideas with zero clutter. The mistake most people make (even seasoned designers!) is not editing enough. Once you achieve elegance in form and function, your content can shine!
Depends a lot on what the site is about. What makes it professional to me is if it is successful at conveying the message. Does a hotel website make you wish to visit there? Does an artist's site make you want to buy a work of art for sale on that site? Does the site make you want to come and look at it again and again? The last part is made up of looks, with a hook, and how is it to navigate? Does it tell a story? Better way to put that is What Story Does It Tell?
1. Site should hook the viewer visually,
2. Simple to Navigate,
3. Tell (sell), a good story.
I think this one does all three:
Christine Gregory Photography