More recently, pizza box has come to refer to a server specifically designed for use in high density rack environments, usually in large numbers. The name comes from the shape of such a server, which is usually wide and deep, but only one rack unit (1.75") high – almost literally the size and shape of a large pizza box.
Pizza boxes are specially adapted for environments with large numbers of identical servers, such as a server farm for web hosting or a Citrix terminal server installation, just to name a few. Due to the boxes' small size, they have very little internal storage capacity, under the assumption that they will be connected to some form of external storage such as NAS or a SAN. Many redundancy features found in larger servers are also sacrificed, again because pizza box-style computers are intended for use in an environment where some other form of software-level fault tolerance, such as clustering or load balancing, is provided. Essentially, a pizza box is an appliance, built to be used in a larger, more sophisticated software and hardware system, and replaced in its entirety if it fails.
As of this writing, popular Intel-based pizza box servers include HP's Compaq Proliant DL320 and DL360 servers, the Dell PowerEdge 350 and 1650, and IBM's xSeries 300, 305, 330 and 335. Apple, not to be outdone, announced their Xserve 1U MacOS X server product in mid-20021. For extremely high-density deployments, some expect the blade server to compete with or even supplant the pizza box, but the technology has yet to be proven.
1: Thanks to cbustapeck for pointing out Apple's Xserve.