The Arduino Diecemilla board uses the Atmega 168 processor, the older boards uses the Atmega 8 processor. These are currently the only processors you can use with Arduino.
It is possible to program the processor and then remove it and run it "stand alone", you will need a few supporting components, the 16Mhz crystal for clocking the processor, and a couple of capacitors if i remember correct. And of course a power supply of some kind.
One Atmega168 processor comes with the arduino board, it has the bootloader already burnt in it. More can be bought many places. You have two choises either to buy additional processors with the bootloader already burnt (about 5$ a piece) or to buy them raw and burn the bootloader yourself, which requires additional hardware.
You could also buy some of the Arduino compatible small footprint boards, like the Really Bare Bones Board from http://www.moderndevice.com/
or the Boarduino from http://www.adafruit.com/index.php?main_page=index&cPath=19
. Both sites also sell lots of other Arduino related stuff including processors with the bootloader.
The small footprint boards needs a special cable because they do not ghave the FTDI (USB to serial) chip on the board. But if you need a few boards / processers this might be a good deal since you only need one cable to program any number of these boards that are cheaper tha the "full" Arduino board.
Hope this helps a little