Most people think of touchscreens when they think of tablets, because this is the interface element most people use to interact with the operating system. The underlying hardware, however, is an important part of how a tablet PC works.
Generally speaking, tablets use an ARM based processor like that found in a smartphone. In fact, the hardware in some tablets is nearly identical to hardware that you would find in smartphones. The current generation of ARM processor hardware usually tops out at a clock speed of 1 GHz, and is typically single core, although some new dual-core variants have been displayed.
This processing hardware is usually paired with Flash memory, like that you would find in a solid state hard drive or a USB drive. This memory is small, runs cool, and has no moving parts, making it perfect for a device that must be thin and light.
There are a handful of tablets on the market that make use of x86 processors from Intel. The upcoming ASUS Eee Slate is one example - its powered by a formidable Core i5 processor. However, most steer away from x86 compatible hardware. Both Intel and AMD are still primarily focused on desktops and laptops, and the power draw of their processors isn't suitable for most tablets.