The term 'Web server' can be used to refer to a software program which serves Web content, or to a computer running such a program.
Web server: software
A Web server program is a piece of software that can handle requests across the Internet for Web pages, graphics, and related files.
Web servers accept requests from clients (Web browser programs running on user computers), and send responses to those requests. A response might be the requested file, or it might be an error message explaining why the server cannot provide the requested file.
Web server: hardware
The term 'Web server' can also be used to refer to the computer which is running the server software. Generally, this computer will have a fast, permanent Internet connection, but more or less any computer can be used. Web serving doesn't require high-performance computers; many people use old hardware such as 486s to serve small Web sites.