In what medium are non-mechanical waves a disturbance? The aether? The most basic and simple definition of a wave is that it is a disturbance in a medium. Then what are non-mechanical waves disturbance in? The aether? Please explain to me this question in detail as simple as possible for I am in high-school.
 A: Your premise is correct. Waves don't have to be a disturbance in a medium. For example, non-mechanical waves, as you have just said.
But it is wrong to assume that it is a disturbance in the aether. They are oscillations of a field. For example, electromagnetic waves are disturbances in the electric and magnetic fields (as the two are perpendicular, the oscillations are too. With a Fourier transform, we can show that they are sine waves).
I hope it helps. I'm new to SE, so I'm not sure how much is enough.
A: It sounds like a simple question. But thinking about it carefully and making careful measurements lead in very unexpected directions, such as Special Relativity. 
Let's start with an electron. Call it A. It is charged, and so it exerts a force on any other nearby electron, B. We leave A at a fixed location. The magnitude and direction of the force on B depends on where B is. 
Given any location x, we can calculate the force that B would feel if it was at x. A fills all space with the possibility of a force on B. We can can calculate the would-be force without B being present. We say that A fills all space with a force field. A force field is a mathematical abstraction that tells us how forces work. There may be nothing but vacuum around A. 
An electric field is very much like a force field. The force on B is proportional to the charge on B. We want something that only depends on A. So we divide the force by $q_B$. That is the electric field created by A. 
It turns out that the electric field is very useful tool, especially when charges move. Moving charges also create a magnetic field. We will leave it aside, as it is much like the electric field. 
Suppose A moves back and forth, while B is fixed. Then the force on B changes as A moves. But B does not feel the force change right away. The "news" that A has moved travels at the speed of light. 
Mathematically, it is straightforward to describe oscillations of the electric field traveling at the speed of light. When the oscillations get to B, they tell you about the forces on B. 
It is very easy to think of the electric field as a thing instead of a mathematical abstraction. After all, it moves and carries information. There is such a strong parallel between the abstract field and the physical forces on electrons, that people think of them as the same thing. You can see that happening in this post. Is it correct to say that electromagnetic waves does not require a medium? And this is fine. It works. 
But it does get confusing when you ask what is waving. 
In the 1800's, physicists expected that if electric fields have waves, some substance must carry those waves. They named the substance aether, and tried to work out what the its properties must be. 
The properties were strange. Aether permeates everything. It has no mass. And so on. Strange, counter-intuitive properties are fine. Physicists today believe in very strange properties, and do experiments to show they are correct. 
The idea ran into trouble around 1900. As Earth orbits the sun, it approaches a star and then recedes from it. If light is an oscillating electromagnetic field that travels through aether, we should measure a higher speed when we approach the star. Michelson and Morley tried to measure the difference in speed, and could not do so. Various stratagems were tried to explain things. Nothing worked until Special Relativity explained it without aether. 
Today we say nothing is waving. 
