For a Standalone Installation

Processor: of 1ghz or above
Memory: 512MB of RAM (1GB recommended)
Disk: 20GB of disk space depending on options
Graphics: If you plan to use the netbook interface, you will require a graphics card that supports 3D acceleration on Ubuntu. Most netbooks are supported very well and shouldn't be problematic.
If you don't understand any of the above: Most computers made in the last 3 years should be fine, and many older computers too.

When Installing an LTSP Server

Processor: Core 2 Duo or Core 2 Quad as recommended minimum. Simpler CPUs will work, but at a performance penalty.
Memory: 512MB for the server itself, and between 256MB and 768MB of RAM per client, depending on the demands of the users. Typically, you would want at least 4GB RAM for 20 LTSP users, and ideally at least 8GB. Note that you can also use local applications (called local apps) to shed some memory usage to the thin clients.
Disk: 20GB Depending on options. It's recommended that you keep the /home partition on fast disks for optimal performance, refer to the post-install section in this document for more information on setting up a home partition on RAID.
Network: LTSP can be very demanding on local network resources. It is recommended that you have at least 1 gigabit connection to the server for every 15 thin clients, and at least 100mbit connections to the thin clients from your switches.

LTSP Thin Clients
LTSP allows you to set up a diskless environment. If you don't want to set up an LTSP server, then you can skip this section.

CPU: Intel i686 compatible CPU. Most processors will work just fine. Geode and older Via CPUs are not supported.
RAM: 128MB minimum, more RAM is required for running local applications.
Networking: 100mbps network card that supports PXE or similar netbooting.