The computers in question also need to meet certain memory and performance standards. Canonical's recommendations are 512MB - 2GB of memory and 40GB - 200GB for the controller, and 1 - 4GB of RAM and 40 - 100GB of storage for the nodes. It probably goes without saying that you'll want the fastest possible network connection between all elements: gigabit Ethernet whenever possible, since you'll be moving hundreds of megabytes of data among the nodes, the controller, and the outside.
Finally, you'll need at least one machine that can talk to the server over an HTTP connection to gain access to Eucalyptus's admin console. This can be any machine that has a Web browser, but for the initial stage of the setup it's probably easiest to use an Ubuntu desktop client system. The scripts and other client-end technology provided all run on Ubuntu anyway, so that may be the best way to at least get the cloud's basic functionality up and running.
Setting up a UEC instance is a bit more involved than just setting up a cluster, although some of the same procedures apply. It's also not a good idea to dive into this without some existing understanding of Linux (obviously), Ubuntu specifically (naturally) and cloud computing concepts in general, as well.
The first thing to establish before touching any hardware is the network topology. All the machines in question -- controller, nodes, and client-access machines -- should be able to see each other on the same network segment. Canonical advises against allowing another DHCP server to assign these machines addresses since the node controller can handle that duty. I've found you can get away with having another DHCP server, as long as the IP assignments are consistent (e.g., the same MAC is consistently given the same IP address).
The actual OS installation is extremely simple. Download the Ubuntu Server 9.10 installation media, burn it to disc or place it on a flash drive (the latter is markedly faster), boot it, and at the installation menu select "Install Ubuntu Enterprise Cloud." You'll be prompted during the setup for a "Cloud installation mode"; select "Cluster" for the cloud controller. (You can optionally enter a list of IP addresses to be allocated for node instances.)
After the cluster controller is up and running, set up the nodes in the same way. At the "Cloud installation mode" menu it should autodetect the presence of the cloud controller and select "Node." If it doesn't do this, that's the first sign something went wrong -- odds are for whatever reason the machines can't see each other on the network, and you should do some troubleshooting in that regard before moving on.
Once all the nodes are in place, you'll need to run the euca_conf command on the node controller. This insures that the controller can see each node, and allow it to be added to the list of nodes available. The controller doesn't automatically discover and add nodes; this is a manual process for the sake of security. (If you have more than one cluster on a single network segment, you can quickly see why this is a smart idea.)