What is the amplitude of a wave? What does it represent in sound waves? Electro-magnetic waves? Light waves?

First, know that light waves are electro magnetic waves; so the answer will apply to both. I think your fessor is trying to trick you with this question. Both are transverse, S waves. They oscillate perpendicularly to their direction of travel. Sound though is a compression, P wave. It oscillates in the direction of travel; i.e., fore and aft along the direction the wave is going. Amplitude is a measure of the distance between some average or neutral level and the maximum or minimum level that each wave reaches within one cycle. For P waves, that means the 1/4 wave length distance between max compression or max rarifaction and the average or neutral (ambient pressure) point. For S waves, that's the distance between the crest or trough, and the average or neutral point. In both S and P waves, the intensity of the wave is proportional to the square of the amplitude. For EM/light that's the brightness or magnitude; for the P wave that's loudness or magnitude. BTW: Amplitude is not a measure of energy, but it is a measure of how much total energy the wave can carry... the intensity. For example, the energy of a single photons is e = hf; where f is the frequency and h is a constant. So EM or light energy is proportional to the frequency. But the total energy per unit time in a beam of light, for example, would be E = Ne; where N is the number of photons and that number is determined by the amplitude.

There is some ambiguity in the definition of amplitude, whether it means peak amplitude, peak-to-peak amplitude, or RMS amplitude. To be clear, if you have a wave equation y = A sin (kx - wt) then A is the peak amplitude or half the peak-to-peak amplitude. Most physical waves consist of two oscillating variables. You can measure the amplitude in terms of either one. For sound waves, the peak-to-peak amplitude would be the difference between the maximum pressure and minimum pressure in the air (or other medium) the sound is passing through. Alternatively, it could be measured as the difference between the maximum and minimum velocity of the air particles. For electromagnetic waves, it's the difference between the maximum electric field and the minimum electric field, or alternatively the difference between the maximum magnetic field and the minimum magnetic field. They will be related to each other by a constant factor anyway. Light waves are electromagnetic waves, so same answer for them.

There is some ambiguity in the definition of amplitude, whether it means peak amplitude, peak-to-peak amplitude, or RMS amplitude. To be clear, if you have a wave equation y = A sin (kx - wt) then A is the peak amplitude or half the peak-to-peak amplitude. Most physical waves consist of two oscillating variables. You can measure the amplitude in terms of either one. For sound waves, the peak-to-peak amplitude would be the difference between the maximum pressure and minimum pressure in the air (or other medium) the sound is passing through. Alternatively, it could be measured as the difference between the maximum and minimum velocity of the air particles. For electromagnetic waves, it's the difference between the maximum electric field and the minimum electric field, or alternatively the difference between the maximum magnetic field and the minimum magnetic field. They will be related to each other by a constant factor anyway. Light waves are electromagnetic waves, so same answer for them.

Think of amplitude as maximum disturbance of "medium" from the undisturbed state. In the simple example of transverse wave on a string, it is the displacement from the line of the undisturbed string. In the case of sound waves, it is EITHER the pressure variation from the free stream fluid, OR it is the (statistical) maximum displacement of any given group of fluid from equilibrium in order to cause the compressions and rarefactions. These two amplitudes are related, and also depend on wavespeed and frequency. In the case of light and electromagnetic waves, it is the maximum variation in value of Electric field OR Magnetic field. These two "different" amplitudes aren't really that different. They are INHERENTLY COUPLED via Electric amplitude = magnetic amplitude * speed of light. Yes, I KNOW that EM waves have no real medium, but for your immediate purposes, accept that the "medium" is the electromagnetic field effects.

Think of amplitude as maximum disturbance of "medium" from the undisturbed state. In the simple example of transverse wave on a string, it is the displacement from the line of the undisturbed string. In the case of sound waves, it is EITHER the pressure variation from the free stream fluid, OR it is the (statistical) maximum displacement of any given group of fluid from equilibrium in order to cause the compressions and rarefactions. These two amplitudes are related, and also depend on wavespeed and frequency. In the case of light and electromagnetic waves, it is the maximum variation in value of Electric field OR Magnetic field. These two "different" amplitudes aren't really that different. They are INHERENTLY COUPLED via Electric amplitude = magnetic amplitude * speed of light. Yes, I KNOW that EM waves have no real medium, but for your immediate purposes, accept that the "medium" is the electromagnetic field effects.