First lets unpackage the AirPort Express and plug it into a power outlet. For this tutorial our network will look similar to the image below. (Old image but configuration still applies)
Give the Base Station a few minutes to boot. Then click the AirPort icon at the top right hand corner of the screen. Select the Apple Network from the dropdown.
Open a new Finder window and run AirPort Utility from within Applications : Utilities.
If you do not have AirPort Utility you will have to install it from the CD that came with your AirPort Express. Insert the CD and a window will popup showing the contents of the disc. Double click the AirPort Express package icon begin the installation. Once installation is complete open a new Finder window and run AirPort Utility from within Applications : Utilities.
Airport Utility will search and find the connected Airport Express. Click the Continue button to set up your base station.
To personalize your AirPort Express name your Base Station and give it a password. Then click the Continue button to continue setup.
Now you must choose the type of setup for this AirPort Express. Select the option I want AirPort Express to wirelessly join my current network. Then click the Continue button.
From the Network Name dropdown choose the wireless network you would like to extend. From the Wireless Security dropdown select your networks security type and then enter your username and password. Finally make sure you check Extend the range of this network. Click the Continue button when done.
The AirPort Express will then be configured to join your existing wireless network.
A popup will appear. From this popup select the Apple wireless device you want to extend. Then click the OK button.
Your existing network will now be configured to allow it to be joined.
Congratulations your AirPort Express is now configured to extend your AirPort Wireless Network.