This is the LilyPad Arduino Simple Board. It's controlled by an ATmega328 with the Arduino bootloader. It has fewer pins than the LilyPad Arduino Main Board, a built in power supply socket, and an on/off switch.
Magnets. How do they work? We're going to guess magic. This particular set of magnets are pretty darn strong, are able to rotate freely, and conduct electricity. That's a pretty darn good kind of magic, if you ask me.
This is a large board for prototyping through hole components with the LilyPad system. This allows for quick and easy prototyping - all you have to do is cut the traces between holes you don't want connected!
The LilyPad Simple Power is a board that lets you connect a battery (or other power source with a 2-pin JST connector) and switch it on or off, that's it!