In a restaurant, a client sits down to order a meal and the server takes the order and delivers the meal. So too, in a way, with servers and clients on the web. Your browser is a client. Websites are stored on a computer somewhere (at least one) and served to your browser when you type in the correct URL (Uniform Resource Locator). An example of a url is http://www.sherwinarnott.org/.
Every site, every document, every location on the web has a unique address or Uniform Resource Locator (URL), much like every file on your computer has a unique address. Much the same way your computer might get a little slow if you were trying to access a lot of files all at once, servers can get overwhelmed if your site is busy and the server has not committed enough memory and processing power to your portion of the server.
Where personal computers are computers designed to be used by a single user, servers are pretty much computers designed to be accessed by many, even thousands, of users. So while it’s possible to turn a personal computer into a server and serve your website to browsers on the internet, from your home computer, it’s also fun and possible to hire someone who specializes in this sort of thing to do this for you.