There are so many types of servers and even I, a network engineer could not name them all.
But I will name the most common, as well as what a server is and how they work in this article. So keep reading.
What is a Server
A server is a type of computer or program that delivers data to another device. Either over the internet or a local network. Server computers have special programs and hardware to be able to perform their task efficiently. they use special operating systems, for example.
The server receives requests from other devices that want information. The server stores this information and sends it to the device requesting it. In this system, the server is the host, and the devices that are requesting are the clients. 
What Servers Do
Servers can do a lot of different things, but the essential is that they are providers of information. Most modern servers are specialists and only perform one task. 
These tasks can for example store websites as web servers do. This allows you to write in the website domain in your browser and then receive the website to your browser from a web server.
How Servers Work
To understand how a server operates I will explain with an example. You write a web address (like youtube.com for example) into your browser and hit enter.
This will first convert the web address into an IP using a DNS server and then get sent to the Web server storing the site.
The web server will receive your request via a port and send back the website information. Your browser will use this information to construct the website.
Because servers take thousands of requests a minute they use specialized hardware and operating systems. And since they are never turned off, this hardware must always work.
For example, servers use a specialized system of disks called RAID (Redundant Array Of Inexpensive Disks). This works by having many hard drives storing the same information. So, if one breaks the server use another disk. 
Furthermore, Disks are always extremely easy to replace. This makes RAID is very important. After all, servers need to work all the time because so much is dependent on them. A web server for example must keep its website online.
In addition to using RAID, servers have tons of RAM and many CPUs. This allows the server to handle thousands of requests for data from all over the world.
Physical & Virtual Servers
A physical server is a computer that runs the server. This could be a home computer or a specialist server computer.
A virtual server is a server that shares resources with many users that have control of it. The consequence is that it is possible to convert one physical server into multiple different virtual servers. This helps to lower the hardware and energy cost. 
Types Of Servers
There are a lot of different servers, here are the most common ones.
This server’s role is to watch systems and networks. Sometimes they receive the client’s request but don’t answer them themselves. This can be to keep track of traffic in a network. Perhaps to keep statistics of the number of visitors to a site. Monitoring servers also make sure everything in a network is working. This helps the administrators to keep the network healthy.
Application servers run resource-heavy applications that many users constantly are using. This makes application servers use very strong CPU and RAM hardware. These kinds of servers often run applications like web applications and hypervisors.
These kinds of servers store and distribute data over the internet or the local network. File servers are often used in large organizations because it offers a secure centralized solution for file storages.
A web server is what delivers you this web site. It stores websites on it and when you type the web address into the browser. It will send a request to the Web server. This is one of the most common types of servers today. And with over 1 billion sites it is understandable.
A blade server is a small electric circuit board, each one being a server. They are often dedicated to a single task. They use as little physical space and energy as possible.
A Cloud server is a service through the internet. Instead of being one server, a cloud server is a complex of different servers connected over the internet. They are often hired by companies that don’t want to buy and maintain their own server array.
Domain Name System (DNS) are servers that are responsible for converting the domain name (web address) of a site into an IP address that computers can understand. This is because domain names are human-friendly but computers don’t understand them.
A proxy server sits between the client application and the real server. Its purpose is to filter requests (mostly from bots) and improve performance. The most common proxy server is the one sitting between web servers and web browsers.
Database server performances a lot of tasks. But first and foremost they work with data. They can for example store, analyze, manipulate, or archive data. They are accessible for many clients and need insane amounts of disk space. Famous database server providers are Dropbox and Box.
Instead of placing a server in every printer, you can use print servers. This is a server that receives all the printer requests from clients. And distributes them to the right printer.
Mail servers are what handles, delivers, and store emails. They receive the emails from a client device and then send them to their intended destination. They also store all the emails, which is what allows you to log in to any device and still see all your old and new emails.
A server is what is the reason you can read this, it is what delivered this information to your browser. Servers are the basis of the internet and are distributors of information.
If you want to learn more about how the internet works check out what’s a router and how it works.