Windows Home Server comes with Connector software that is installed on each client machine that you wish to back up.
There is a 10 client machine limit for Windows Home Server, meaning that you can have 10 machines set to backup automatically at any one time.
Windows Home Server comes with client install DVDs that can be used to install the client, but in general, once WHS is installed, it is better to install the client software from the shared folder. For example:

Here I have my Windows XP laptop browsing to the \\server\Software folder. It is a convenient place to store the connector and (as you can see) Restore software so that you don't have to carry a DVD with you as you set up your network.
Inside this folder you will have the setup.exe and .msi files for x32 and x64 bit installations.
PLEASE NOTE: prior to Power Pack 1 for Windows Home Server, 64 bit client software was not included. If you have a Home Server and are looking for the 64 bit client support, it is available by installing Power Pack 1 (KB944289) from here.
Run setup.exe from \\server\Software\Home Server Connector Software. This will guide you throught the setup process for each machine.

Windows Home Server will prompt you for the WHS Admin password. This lets WHS know that the client machine is allowed to be part of the WHS network.

In most cases, everything will be fine (provided you entered the correct password). However, if you encounter an error such as the one below, please see the Windows Home Server Troubleshooting Guide.

But luckily, most, if not all of you, will get the following, happier message:

I recommend choosing to download the updates from my home server and install them automatically. There are almost no cases (except when investigating a problem) that you would want to spend the time and manually install updates. But of course the choice is up to you.
Windows Home Server can even wake up your sleeping computers to back them up. This bit of magic requires that your computer has Wake on Lan enabled in the BIOS and almost certainly requires that it is connected via ethernet and not wireless.

After this choice is made, your client joins the Home Server and configures the backup.

At this point, you've completed the client install for WHS. Next, you'll want to connect to the Windows Home Server console and configure settings like when you'd like backups to happen, etc.

Next: Windows Home Server Console