You Will Need
Extract the the zip file and make a note of where the extracted folder is on your file system.
Copy NOOBS onto an SD Card
The archive will extract to a folder called _NOOBS_v1_3_1_ or similar, it is the contents of the folder that should be copied to the SD card, not the folder itself.
The SD card should be a new SD card, or one that has just been formatted as FAT. If it's a new SD card, its ready to go. If you are recycling an SD card, reformat it so you get the most space available with the official SD association formatting software
Simply copy the contents of the downloaded and extracted NOOBS folder onto the SD card. This will create an SD card from which the Raspberry Pi can boot.
Boot your Pi
When it boots, the window shown below will appear. From this screen, you can select which distribution you want to install. Raspbian will be selected as the default, and this is the best choice to start off with.
If you selected Raspbian (and some of the other distributions) the Raspi-Config untility will automatically run so that you can configure the new installation (see this tutorial).
If you are re-using an old card and need to format it as FAT, then use your operating system's tool for formatting removeable media.
You can also use the SD Association's Formatting Tool, which is available for Mac and Windows and can be downloaded from here: https://www.sdcard.org/downloads/formatter_4/
You can find out more about NOOBS here.
About the Author.
As well as contributing lots of tutorials about Raspberry Pi, Arduino and now BeagleBone Black, Simon Monk writes books about open source hardware. You will find his books for sale here at Adafruit.
His latest book: "The Raspberry Pi Cookbook" is out December 2013.