Browsers use hypertext transfer protocol (HTTP) to communicate with web servers to request content. Once the browser receives the content back from the server it uses its internal rendering engine to translate and display/render the content on screen.

Browsers are made up of a User interface (Address bar, visual elements etc), Internal Engine, Rendering engine (parse HTML & CSS), Networking technologies, Data Storage (including session management) and a Javascript interpreter.

With each browser having all these unique parts it explains why different browsers display differently, with the rendering engine being the most important difference.

Some handy links I came across while writing this

taligarsiel.com/Projects/howbrowserswork1.htm
www.html5rocks.com/webappfieldguide/toc/index/
www.20thingsilearned.com/en-GB/home
www.w3.org/