What exactly are the X and the Y axis of a brainwave?
In the general case of EEG, the x-axis is time, and the y-axis is the voltage potential measured by an electrode placed on the scalp.

When others talk about "brain waves" they typically refer to oscillations visible in EEG electrode signals that have more power in a certain frequency range. For example, "Alpha brain waves" usually refer to EEG signals with maximum power in the range of 7-13 Hz. "Gamma waves" have frequencies >30 Hz.

The dominant theory of the origin of the waves is that they are the result of large populations of individual neurons firing action potentials in a synchronized fashion. 

So, if you had the ability to make, say, a 1,000 neurons near the surface of the scalp fire at the same time, a scalp electrode placed above those neurons should register a significant voltage potential deviation around the time of the trigger.