Confused about what a wave is When a wave of something, let's say light or some electromagnetic wave is given, I am confused because I do not understand if shape of a wave represents projectile of it or some value that possess at certain positions. I researched a lot but I have no idea what a wave really is. 
My question is: what is a wave?; what defines a wave?
 A: 
when a wave of something, let's say light or some electromagnetic wave is given,

You have chosen a wrong "say" example, because light is not  a typical wave. Sound waves, water waves obey a wave equation that describes the transport of energy by displacements in (x,y,z) at time t . The simplest solutions of wave equations are sinusoidal. 
It all depends on what physical  meaning you give to the variables in the wave equation. ( for example , in quantum mechanics the solutions of the wave equations end up as describing probability of interaction, another story)

I am confused because I do not understand if shape of a wave represents projectile of it or some value that possess at certain positions.

It is values represented in certain (x,y,z) at time t.
It was thought that electromagnetic waves were the same type of waves, energy moving on a medium, in Maxwell's equation represented by electric and magnetic fields. The Michelson Morley  experiment showed that electromagnetic waves were self propelled , not needing a medium to transport their energy. So it is energy transported by sinusoidal solutions , as described in this link. This energy  transport has a direction, and is called the Poynting vector.
So a wave in physics is modeled by the wave equations that have sinusoidal solutions, and are used in various frames to model observations.
A: If you're trying to understand the idea of a wave, I recommend that you start with a simple example: a transverse wave on a long string kept under tension. I'm afraid that you must draw your own diagrams.
Suppose that you (acting as wave source) displace one end of the string transversely. The displaced portion will exert a transverse force on the neighbouring of string, which will become displaced and exert a transverse force on the next portion and so on. In this way a transverse wave will travel along the string. [We won't worry about what happens when the wave reaches the other end.]
A more typical wave source would be continuously oscillating, so a whole sequence of displacements (maybe up and down) would travel along the string. We are now in a position to answer your question...
There are two sorts of graph that you are like to meet. 
(a) One is a displacement-time graph for one chosen particle on the string (or in the path of the wave). If we draw such graphs for two particles, the oscillations will generally be out of phase, because the wave takes longer to reach the particle further from the source.
(b) The other sort of graph is a 'snap-shot' of the whole wave, that is a displacement–distance-from-source graph drawn at one particular time. [It will look very similar to the displacement-time graph for a particle, but the interpretation is, of course, quite different.] If we draw another snapshot graph (displacement–distance-from-source) for a slightly later time, we would see that the whole wave has advanced away from the source.
You can draw exactly the same sorts of graph for an electromagnetic wave. For displacement you plot the electric field strength vector, which is transverse to the direction of travel. 
A: You are asking whether the shape of the EM wave represents a projectile (projection) of it, or some value that the EM wave posesses at certain points.
You are confused because you see these images of EM waves. 

In reality, it is the value of the EM field at any given point in space that these diagrams represent for the electric and magnetic field.

In light propagation, oscillation does not mean any movement in space. It is the value of the electromagnetic field, at one given point in space, that oscillates. For electromagnetic waves, there is no matter or photons that go up and down. Instead, you have to imagine that there is a little arrow associated to each point in space: this little arrow is the electric field direction. Another arrow, at the same point, is the magnetic field. These two arrows change size and direction with time, and in fact they oscillate.

How does light oscillate?
A: The “shape” of the wave is exactly what a wave is. It is a “shape” formed in a medium in which it moves while maintaining the shape (ideally). The shape formed is a property of the medium and of the object that caused the shape to form in the medium. 
For example think of the medium as water in a pond and the object as a stone dropped in it. This causes the water around it to ripple and form a shape that moves radially outwards. This is what a wave is. At least one kind of wave. And since it is only dependent on the medium and the source, we can describe the entire thing in terms of a relation called the wave equation. 
As for light, the medium turns out to be something called as the Electromagnetic Field and the source is movement of charged particle.
A: The most basic definition of a wave might be $f(x-ct)$ which is the equation of a function that moves to the right (translates) with a speed $c$. $f(x-ct)+g(x+ct)$ is the equation of two waves--one moving to the right and one moving to the left--and this equation is the general solution to the 1D wave equation.
So the most basic elementary waves are disturbances that move in space at some constant speed without changing their basic shape. (In 3D the waves are attenuated by spherical spreading but still retain their shape) 
