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Breaking any encryption involves knowing a few things. First, you have to know that there is an encryption scheme. Secondly, you must know how encryption works.

[edit] Steps

First you will need a to get a packet sniffing program compatible with your Operating System.
Some packet sniffing programs are kismet, Airsnort (sniffs and cracks WEP), and airodump-ng(included in the aircrack-ng). I would recommend using aircrack-ng.
Set up a wireless access point to use in this experiment, preferably one that you created. Otherwise, you will need the network administrator's agreement.
Allow the packet sniffer some time to get a good sampling of the WEP encrypted packets. In other words, you have to give the program something to work with and match. This may take some time depending on your wireless card. Aircrack-ng has a program to speed this up by doing packet injection.
Decipher the Public/private key by matching their packets for similarities.

[edit] Tips

Many sniffing programs such as the above-mentioned Wireshark as well as Airsnort and Kismet are available as source code. You will need some background in compiling source code for Linux or Windows to use Airsnort or Kismet. Ethereal comes with an installer or you can download the source code.
You will probably be able to find already compiled versions of most programs you will need.
Laws concerning network security can differ from place to place. Make sure you know everything you need to know and be prepared to face the consequences of your actions while attempting this.
Guides to help you crack WEP keys:

smallnetbuilder.com's "WEP Cracking...Reloaded" Guide
airdump.net's "Hacking WiFi - Ultimate Ubuntu Guide"

[edit] Warnings

This information is to be used ethically. Misuse of this information may be illegal both locally and federally.
You need a specific wifi card that is compatible with the programs you are using
Always be careful of who your target is. It's not smart to walk into your local McDonalds and try and crack their net. The chances of you getting caught increase ten fold.