The Web Server Makes It Public

The Web server is a software application that exposes a Web site for others to see. You can't have a Web site without a Web server, because the server is what a browser communicates with in order to fetch Web pages. Without the server, your Web site is just a bunch of files sitting in your computer that only you can access.
Unlike a Web site, a Web server is complicated to set up and (especially) to maintain. In fact, there are all kinds of reasons not to run the Web server software yourself, even if you're technically inclined. I'll get to that later, but consider this: If I don't run a Web server, why should you? Let someone else run the Web server for you on their own computer. This is called Web hosting and it'll get your site up and running a lot faster than setting up your own Web server.
Web Server Confusion
Sometimes Web server refers to the physical computer that the Web server software runs on, as opposed to the software itself. To avoid this confusion, I always say Web server to refer to the software and server computer to refer to the computer on which the Web server runs.