After a couple hours of searching top lists of free "web servers" and "web frameworks" on google, I realized I'm not sure what I'm looking at.
Let's take a couple of example softwares I have come across :
IIS and Apache are web servers that support a variety of different technologies for plugging code into the web servers. So, if you were using one of these web servers, you would also use some other framework and language runtime to plug into them.
Apache Tomcat is a Java-based server framework for web applications. From the wikipedia page: "Apache Tomcat, often referred to as Tomcat, is an open-source Java Servlet Container developed by the Apache Software Foundation (ASF). Tomcat implements several Java EE specifications including Java Servlet, JavaServer Pages (JSP), Java EL, and WebSocket, and provides a "pure Java" HTTP web server environment in which Java code can run." It may be used independently of the Apache Web Server. The "Apache" in the name has to do with it being part of the Apache open-source organization. So, if you want to program your server in Java, this is one of your options.
Django is a python-based framework. If you want to program your web server in Python, this is one of your choices.
socket.io is really not like any of the others. socket.io is a layer on top of webSockets that gives you a whole bunch of additional features over webSockets. This would run in whatever server-side language and framework you were already using and you would have to select a socket.io library that was compatible with your server technology. You can see a list of features that socket.io adds on top of a webSocket here: Moving from socket.io to raw websockets?. webSockets themselves are used for continuous communication between server and client (more efficient than repeated Ajax calls) and for push notification from server to client.
So, if you were looking to select a technology from scratch, there are a number of different ways you could start your selection process.
If you have an existing hosting company that you want to use, you may need to understand which technologies they support so you don't select something that would require you to change hosting companies. Not all hosting companies support all technologies and some are better specialists in some technologies.
If you have specific 3rd party code or libraries you want to use, you may want to investigate which languages/frameworks it can more easily be used with.
What is fresh and what the cool kids are using changes pretty regularly. I would suggest that's a crummy criteria. What makes more sense is to understand which technologies and frameworks are doing well with accelerating growth and a rich development community behind them vs. which ones may be falling out of favor. I haven't myself done any sort of overall survey to offer any sort of list of what's on the upswing and doing well vs. what is more stagnant.
I'm personally partial to node.js and the huge NPM library of open-source and compatible code you can draw from and the ability to use one language for both front-end and back-end. It is certainly one of the choices that is growing rapidly and has a vibrant development community behind it. But, it is not the only choice that would meet those qualifications.