The BCI is a collaboration between a brain and a device that enables signals from the brain to direct some external activity like moving the cursor on a computer screen or to determine the direction a robot moves.
Neurofeedback (EEG biofeedback)
Neurofeedback is direct training of brain function (brainwaves), by which the brain learns to function more efficiently.
By using a EEG device we observe the brain in action from moment to moment. Then we show that information back to the person and reward the brain for changing its own activity to more appropriate patterns.
For instance the brain-alpha-waves correpond to a more quit mind. To stimulate these we could show a figure on a screen which is growing if the alpha-waves get stronger. The person is asked to make the figure bigger and so doing learns to influence his alpha-waves.
Links about neurofeedback
International society for Neurofeedback research
A nice video explaining neurofeedback
What is the EEG (Wiki)
Marty Wuttke Institute
Interviews about new fields in mind research