To put it simply, a microcontroller is something like a very small computer designed to be used in embedded systems. The Arduino board is a small circuit board that features a programmable microcontroller. It's an open source project and comes in many different flavors. There are also options other than Arduino, but since this is so common, and we can't offer a complete microcontroller course here, we'll just stick to Arduino.
While Arduino can act on its own, without a computer controlling it, it's also the most common method of getting miscellaneous (non audio) voltage into or out of Max, TouchDesigner, and the like. This enables us to read sensors, control motors, and build custom MIDI or OSC controllers, and just opens up the wonderful world of physical computing.
In the following screenshot, you can see an Arduino UNO, which is one of the most common Arduino versions:
Original image at http://arduino.cc/en/uploads/Main/ArduinoUno_R3_Front.jpg