When evaluating a web design, this is my personal list of things to check.
This list covers the most important areas, however it's important to remember that there is never one single answer to design questions.
First glance test
In the first 5 seconds, what do I feel? What do I remember?
Does the top fold convey the value proposition?
When a new person visits your frontpage, do they understand what you do?
Tell a clear story
From first arriving on your site to leaving, there should be a logical flow of information to each visitor. Imagine you're talking with the visitor on the phone: would you say the same things as your page does?
Engages visitor to read on
Upon reaching midway or the end of a page, does the visitor have enticing options to continue their stay?
Good information hierarchy
Humans deal with one piece of information at a time. Use clear headers, sub-headers etc. to lead them through the structure.
Drip feed information to the visitor, rather than showing too much to them at once. Dense text is stressful.Navigation is simple and always nearby
Is it easy for visitors to take the next step they wish? Easy at all scroll positions, on all pages? Include navbars, gutter links, frequently offer the most useful next step, consider static elements if vitally important.
Does the site/page have a clear desired outcome?
Usually you want to continue the conversation, or get a signup. Is it always one click away to do so? Can you embed a form in the page to make it fewer clicks?
Is it convenient to take the desired outcome?
When visiting a site the currency is viewers' time. They have a limited budget. Making things more convenient allows visitors to do more in the same budget.
Looks professional / of appropriate quality
Visitors will subconsciously equate the quality of the visual design with the inner quality of the organisation behind the site. You should ensure high enough quality for the situation.
The UI elements follow conventions
UI elements (e.g. links, buttons) look and act certain ways across most websites. If you break these conventions visitors will be confused. Visitors may never realise something is clickable (i.e. it lacks visual affordance), or may fail to use an element properly (e.g. never successfully submit a form with no submit on enter).
Works on desktop, tablet and mobile
An increasing proportion of web traffic is mobile. Mobile is many people's first device for viewing web pages. If you do not work on mobile then you're turning those visitors away
Neat use of design elements
Images should be of good resolution and not overly compressed. Everything should fit a nice simple grid structure (e.g. two columns, horizontal sections). Typography should be readable. Everything should have a good level of polish and neatness.