Web browsers are software you can install on your PC to access the internet. Currently to navigate to my site you must be using some web browser. Browsing or surfing is nothing but navigating through pages of information.
Webserver store HTTP & FTP contents of a website and provides information as per HTTP requests in the form of HTML pages, image files , video files etc. If URL is typed on a browser then the server locates this URL and the html web page is served by the web server.