A Browser is a program that lets you look at and interact with information provided at
World Wide Web Sites. A Web browser uses the Hypertext Transfer Protocol (http) to make requests of these Sites throughout the Internet on your behalf. Generally, you cannot look at information on the World Wide Web without a Browser or a program implementing HTTP as a protocol. In other words, if you can look at a Site, you are most likely using a browser.
The "Client" side of a communication refers to the operations managed by your Browser on your system (Desktop, Mobile Device, Tablet, etc.). The "Server" side of the communication refers to the functions or programs running in the computer that is sending your Browser the HTML files for display.
We highly recommend you make sure your Browser is a recent version (do not permit it to fall into dis-repair as the World Wide Web is a constantly evolving environment). If you are running a less than curret browser, you'll miss out on some cool features, or may not even be able to see a page at all. You can see the current usage of the various browsers courtesy of the Wikipedia folks.