Summary: There are a number of web frameworks in Clojure, but beginners should roll their own server stack themselves to tap into the Ring ecosystem.
One question that I am asked a lot by beginners in Clojure is "What web framework should I use?" This is a good question. In Python, there's Django. In PHP, Drupal. And of course in Ruby, there's the king of all web frameworks, Ruby on Rails.
What framework should you use in Clojure? The question is actually kind of hard to answer. There are a number of web frameworks out there. Some people call Compojure a framework, though it is really a library. lib-noir does a lot of work for you. Then there's your true frameworks, like Pedestal or Hoplon, which provide infrastructure and abstractions for tackling web development. All of these projects are great, but for a beginner, I have to recommend building your own web stack starting with Ring.
Compojure is really just a routing library, not a framework. You can use it for your routing needs, though there are alternatives, such as playnice, bidi, Route One, and gudu. If you don't want to decide, use Compojure. It's widely used and works great. If you want to go in depth, read the docs for the others. They are each good for different cases.
lib-noir comes from Noir, which was a web framework (now deprecated). It was easy and provided a bit of plumbing already built for you, so you could just start a project with a lot of the infrastructure built in. lib-noir is that infrastructure in library form. I haven't used it, but a lot of people like it. However, when I look at it, I see that most of what it provides I either won't use or it is trivial to add myself. That would normally be ok if there was huge adoption for it (like with Rails) so you get an ecosystem effect, but there isn't. lib-noir is used but certainly not dominant.
Pedestal has a lot of backing. It aims to tackle single-page apps by providing a sane front-end environment using ClojureScript in the form of a message queue. If you're into "real-time apps", this may be for you. Though, I would caution you that it's not for a Clojure beginner. Pedestal introduces a lot of new concepts that even experienced Clojure programmers have to learn. The tutorial is long and arduous. You will have problems learning Pedestal without knowing Clojure.
Update: Pedestal has changed dramatically since I last looked at it. The frontend half has been sacked. Pedestal is now just the high-performance backend server for asynchronous processing. It's worth looking at if you need that. Otherwise, I stick with my recommendation to use basic Ring.
Hoplon is also designed for web apps. It gives you a DOM written in ClojureScript (including custom components), dataflow programming (like a spreadsheet), and client-server communication. It's a bold step, but again, requires you to buy into programming models that will take a long time to understand. If you are not already familiar with Clojure, you are asking for trouble.
There are other frameworks out there. But I recommend rolling your own stack. If you're learning Clojure, the best way to grasp how web apps work is to get a Ring Jetty adapter set up with some basic handlers. Add existing middleware as you need it. Write some middleware of your own. Use Compojure to route. Use Hiccup to generate HTML. That setup will get you a long way.
Ring is just functions. With a few basic concepts and a copy of the Ring SPEC handy, you can build a web server very quickly that does exactly what you want and you can understand every aspect of it. The experience of building one yourself can teach you a lot about how the other frameworks are put together.
What's more, Ring is dominant. Most people write functionality (in the form of middleware and handlers) assuming Ring and no more. So by staying close to the metal, you are tapping into a huge resevoir of pre-written libraries that are all compatible with each other. Ring is the locus for the Clojure web ecosystem.
Wiring up your own middleware stack is not that daunting. If you want guidance, my Web Development in Clojure video series is now for sale. It starts with a brand new Clojure project and ends with a fully functional app, backed by a database, and hosted on Heroku (for free!). In one hour, it explains all the concepts and shows you lots of examples. There's lots of exercises to get your brain whirring.
- Ring (with Jetty adapter)
Then keep adding and customizing.