There are many ‘bare bone’ Arduinos you can find from internet. Most of them are on a solderless breadboard with a ATmega168 (or 328) chip with Arduino bootloader, a resonator (with built-in capacitors or needs 2 capacitors), a resistor.Here is a quick drawn Eagle schematic diagram.
This is truely ‘bare bone’ Arduino which means that this simple circuit can do your project if your sketch (Arduino program for your project) is already loaded. If needed be, you can add a reset switch, a power LED, a power converter, an LED for the pin#13 and header pins or receptacles. And most importantly, in order to load your sketches through USB, you will need a FTDI serial to USB converter. So, if you want all of these, the final shape and size will be not very different from the full size Arduino Uno or Arduino Duemilanove. However, if you need a simple device such as a sound sensing light, an LED blinker, a quiz bell (the person who press the button fastest wins), or many more simple (not always necessary to be simple, though) projects, you don’t really need all those accessory parts.