I had a customer ask me this today, so in case others are wondering: What's the “Web Client Network”?
If you open Windows Explorer, click Network Neighborhood, then click Entire Network, you'll see an icon for Web Client Network. The customer saw nothing under this icon, so was wondering why it was there in the UI.
The short answer is that icon is present to display any WebDAV connections that your machine has. As many machines don't make WebDAV connections, that's why there was nothing under the icon for this particular customer.
The long answer: Windows clients can access files on other machines using a variety of network protocols. Each installed network protocol that the client supports gets an icon under Entire Network in My Network Places. The native protocol that Windows clients and Windows servers support is known as SMB. (A version of the SMB protocol was publicly licensed and is called CIFS.) If you expand Microsoft Windows Network under Entire Network, you will see computers that you can connect to using the SMB protocol.
Starting with Windows XP, Windows clients can also connect to machines using the WebDAV protocol. This is a file sharing protocol supported by some web servers, such as http://www.msnusers.com. If you have any active WebDAV connections, you will see them listed in Web Client Network.