Even though your website uses the same files to present your content to visitors, depending on which web browser they use to access your website they might get a drastically different looking website.
While you don't need to create a different website for every different web browser, you should be aware that different browsers can produce different (and often unexpected) versions of your website.
Make sure that as you make any significant changes to your website, you are testing how those changes look in a minimum of Internet Explorer, Firefox, and Google Chrome. Also make sure that any development services you pay for include browser testing and compatibility as part of their service.