The first thing you'll need is the quadcopter itself. The most important component of the quadcopter is the flight controller board. There are several choices that can work, but this example is based on the Hobby King KK2.1.5 flight control board. Here is an image of the board:

The flight control board takes its flight inputs via a set of input signals that would normally come from an RC receiver to control the quadcopter, and then sends out control signals to the four motor controllers. You'll want a flight controller as it has built-in flight sensors, so it can handle the feedback control for the motors. The inputs from the BeagleBone Black will come in as electric commands to turn, bank, go forward, or increase/decrease the motor speed.

You can buy either an assembled quadcopter that uses the HobbyKing 2.1.5 board from HobbyKing at www.hobbyking.com, or you can buy the parts and assemble your own.