Updated Feb, 5th 2013
I type www.google.com into my browser.
My computer sends the request to the Domain Name Server to ask where Google.com is located.
The DNS might not know. If it doesn't know, it asks another DNS, and then another, until it finds the location of Google.com's server. The DNS gets feedback for future reference (if it didn't know where Google is, it knows now, and if it did know it gets confirmation). It's Google, though, so chances are it already knows where it is.
The DNS tells my computer where the Google.com server is located and gives my computer Google.com's IP address. An IP address is the unique "address" made out of numbers, identifying any computer connected to the Internet. (Computers being laptops, routers, servers, my printer, my ipod, and others).
Now my computer knows google.com's address and sends the request directly to google.com's server. Google.com receives my request and responds with data, the page content.
The page content is sent from Google.com's server to my computer in the form of tiny packets. Each packet travels individually across the network to my computer, being routed to me because they are each addressed to me. They contain information about how to assemble them into one form (the webpage on my screen). My computer arranges the packets like a puzzle to make up my webpage.
Now I want to search for funny cat pictures, so I type my seach query, "funny cat pictures" to google.com's server.
Google.com receices my query and searches its database for links containing content that matches my search query. It sends its list of search results back to my computer. My computer receives the data and displays it on my browser. The browser arranges the data into the text and images I see on my screen.