Arduino is a pretty neat idea. It's an open source hardware and software microprocessor environment based on the Atmel AVR® 8-bit series microcontrollers.
The software side of things is programmed in a simplifed C++ programming language derived from Wiring. Your Arduino programs (sketches) are compiled by the Arduino IDE (based on Processing) and then uploaded to the Atmel microcontroller's flash memory. A boot loader allows this to be done easily through either USB or a FTDI TTL-232R USB to 5.0v TTL serial cable (depending on version and/or variant).
The hardware is quite varied. Since the spec is open source, if you can adapt it to a particular function and form factor, and the application is popular, you can produce and sell it. Typically, the major aspect of the board gets 'duino appended to it.
You even can build a uDuino, basically by boardstuffing a solderless breadboard with the correct components, a preprogrammed Atmel microprocessor and applying power.
Arduino is reserved for hardware produced by the official developers. The current board is the Arduino Duemilanove (2009) since upgraded to use an ATMega 328 microprocessor. It can be powered and programmed through its USB port.
The Duemilanove supports expansion boards through a set of header sockets that mate to header pins on the (shield) expansion board. The connections are two 8 pin and two 6 pin sockets. If you use wire wrap sockets in the shield, you can stack several of them for added circuits. Adafruit Industies has a very nicely crafted prototyping shield available in kit form or just the bare boards.
For experimenters who want expansive areas for those larger projects, the Boarduino is a good option. With straight header pins soldered into its peripheral .100 spaced holes, it can be plugged directly into the solderless breadboard. Power it with a 6-9Vdc power supply. If you use a wall wart, it needs a 2.1mm center-positive plug. Programming requires an FTDI TTL-232R USB to 5.0v TTL serial cable.